| L(s) = 1 | − 8·4-s − 20·7-s − 2·9-s + 36·16-s − 4·19-s − 8·25-s + 160·28-s − 4·31-s + 16·36-s + 146·49-s + 40·63-s − 120·64-s + 112·67-s + 32·76-s − 6·81-s + 8·97-s + 64·100-s − 24·109-s − 720·112-s − 114·121-s + 32·124-s + 127-s + 131-s + 80·133-s + 137-s + 139-s − 72·144-s + ⋯ |
| L(s) = 1 | − 4·4-s − 7.55·7-s − 2/3·9-s + 9·16-s − 0.917·19-s − 8/5·25-s + 30.2·28-s − 0.718·31-s + 8/3·36-s + 20.8·49-s + 5.03·63-s − 15·64-s + 13.6·67-s + 3.67·76-s − 2/3·81-s + 0.812·97-s + 32/5·100-s − 2.29·109-s − 68.0·112-s − 10.3·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 6.93·133-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002339019222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002339019222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 + 2 T^{2} + 10 T^{4} + 14 p T^{6} + 82 T^{8} + 14 p^{3} T^{10} + 10 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T^{2} )^{8} \) |
| 31 | \( ( 1 + 2 T - 48 T^{2} + 34 T^{3} + 2334 T^{4} + 34 p T^{5} - 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| good | 7 | \( ( 1 + 5 T + 26 T^{2} + 13 p T^{3} + 274 T^{4} + 13 p^{2} T^{5} + 26 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 + 57 T^{2} + 1600 T^{4} + 29169 T^{6} + 377034 T^{8} + 29169 p^{2} T^{10} + 1600 p^{4} T^{12} + 57 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 18 T^{2} + 770 T^{4} - 9258 T^{6} + 203122 T^{8} - 9258 p^{2} T^{10} + 770 p^{4} T^{12} - 18 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 86 T^{2} + 3320 T^{4} + 80106 T^{6} + 1482894 T^{8} + 80106 p^{2} T^{10} + 3320 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + T + 46 T^{2} - 7 T^{3} + 994 T^{4} - 7 p T^{5} + 46 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 123 T^{2} + 7204 T^{4} + 268253 T^{6} + 7155302 T^{8} + 268253 p^{2} T^{10} + 7204 p^{4} T^{12} + 123 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 96 T^{2} + 4614 T^{4} + 144312 T^{6} + 4001378 T^{8} + 144312 p^{2} T^{10} + 4614 p^{4} T^{12} + 96 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 270 T^{2} + 32786 T^{4} - 2335350 T^{6} + 106652914 T^{8} - 2335350 p^{2} T^{10} + 32786 p^{4} T^{12} - 270 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 132 T^{2} + 11912 T^{4} - 702092 T^{6} + 33511758 T^{8} - 702092 p^{2} T^{10} + 11912 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 99 T^{2} + 3350 T^{4} + 23021 T^{6} - 4475286 T^{8} + 23021 p^{2} T^{10} + 3350 p^{4} T^{12} - 99 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 200 T^{2} + 19280 T^{4} - 1232664 T^{6} + 62511390 T^{8} - 1232664 p^{2} T^{10} + 19280 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 + 147 T^{2} + 14146 T^{4} + 1031891 T^{6} + 57906698 T^{8} + 1031891 p^{2} T^{10} + 14146 p^{4} T^{12} + 147 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 148 T^{2} + 16116 T^{4} - 1233932 T^{6} + 1427458 p T^{8} - 1233932 p^{2} T^{10} + 16116 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 264 T^{2} + 39302 T^{4} - 3869232 T^{6} + 276243490 T^{8} - 3869232 p^{2} T^{10} + 39302 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 28 T + 452 T^{2} - 5340 T^{3} + 48774 T^{4} - 5340 p T^{5} + 452 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 359 T^{2} + 66970 T^{4} - 8048457 T^{6} + 677410634 T^{8} - 8048457 p^{2} T^{10} + 66970 p^{4} T^{12} - 359 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 123 T^{2} + 18040 T^{4} - 1558189 T^{6} + 137107070 T^{8} - 1558189 p^{2} T^{10} + 18040 p^{4} T^{12} - 123 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 409 T^{2} + 80798 T^{4} - 10237143 T^{6} + 935502882 T^{8} - 10237143 p^{2} T^{10} + 80798 p^{4} T^{12} - 409 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 642 T^{2} + 182090 T^{4} + 29797482 T^{6} + 3076803922 T^{8} + 29797482 p^{2} T^{10} + 182090 p^{4} T^{12} + 642 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 377 T^{2} + 79610 T^{4} + 11278983 T^{6} + 1171758954 T^{8} + 11278983 p^{2} T^{10} + 79610 p^{4} T^{12} + 377 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 2 T + 114 T^{2} + 806 T^{3} + 1554 T^{4} + 806 p T^{5} + 114 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.71943531489524008864773601692, −2.62395659035950811900704621461, −2.58873392691082040626790815292, −2.54888188947719302610483510156, −2.25460051376041230869463115610, −2.22521436583151046342049278271, −2.18842760352714814446544644175, −2.09935842095465783821110011997, −2.09502753681610111135441342168, −2.00127677683135219353384667960, −1.93051901291419506064590175078, −1.53968369823339887081159236333, −1.47435880814629783597474408557, −1.45546909627345009869736826255, −1.45023260639379969472640372323, −1.14487671987483780490306549671, −1.10192895873830750144237681807, −0.890133838587161449379217408865, −0.854881806021379094059814245887, −0.75875257530681558189913229083, −0.59816173898315033292937034936, −0.32712134098305778662505573679, −0.16995878354061720977922888542, −0.12730051645729897001040767669, −0.082246094085917184323013618597,
0.082246094085917184323013618597, 0.12730051645729897001040767669, 0.16995878354061720977922888542, 0.32712134098305778662505573679, 0.59816173898315033292937034936, 0.75875257530681558189913229083, 0.854881806021379094059814245887, 0.890133838587161449379217408865, 1.10192895873830750144237681807, 1.14487671987483780490306549671, 1.45023260639379969472640372323, 1.45546909627345009869736826255, 1.47435880814629783597474408557, 1.53968369823339887081159236333, 1.93051901291419506064590175078, 2.00127677683135219353384667960, 2.09502753681610111135441342168, 2.09935842095465783821110011997, 2.18842760352714814446544644175, 2.22521436583151046342049278271, 2.25460051376041230869463115610, 2.54888188947719302610483510156, 2.58873392691082040626790815292, 2.62395659035950811900704621461, 2.71943531489524008864773601692
Plot not available for L-functions of degree greater than 10.
