| L(s) = 1 | − 8·9-s − 32·19-s + 8·25-s − 48·49-s + 20·81-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 80·169-s + 256·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 64·225-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 7.34·19-s + 8/5·25-s − 6.85·49-s + 20/9·81-s + 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 6.15·169-s + 19.5·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 4.26·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1025065329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1025065329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 3 | \( ( 1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 4 T^{2} + 22 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| good | 7 | \( ( 1 + 12 T^{2} + 18 p T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 20 T^{2} + 310 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 20 T^{2} + 406 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 + 52 T^{2} + 1222 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 23 | \( ( 1 - 84 T^{2} + 2814 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 + 76 T^{2} + 2838 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 + 84 T^{2} + 3702 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 84 T^{2} + 3974 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 60 T^{2} + 4206 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 - 180 T^{2} + 12510 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 - 68 T^{2} + 4182 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 212 T^{2} + 18166 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{8} \) |
| 67 | \( ( 1 - 252 T^{2} + 24846 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 92 T^{2} + 10150 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 228 T^{2} + 23526 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - 44 T^{2} - 5466 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 196 T^{2} + 22990 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 260 T^{2} + 32230 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 260 T^{2} + 32518 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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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
−4.26023846643999984491931086043, −4.11693125411179153206936073441, −4.08253443869356656093441159910, −3.82837918064377410292606390095, −3.54179762518627065481637119395, −3.50207224093443734710015805363, −3.48502519311608542932796251612, −3.46866357297740168708394365089, −3.24691575059164648916771440846, −3.22249956800353534867804659320, −3.06203263807475550417502343270, −2.97194048793232155159349602487, −2.82304005995032521481622582497, −2.78618473632705026028466914791, −2.55484236272690200111100406747, −2.48009833454209043096434564647, −2.12174215673087104247097525244, −2.10943270936933276531839013596, −2.07608496722704828197574997567, −1.96727232681080610180130496397, −1.82041611211165762078027645365, −1.73610263099803189247974954658, −1.40528954866311429447308415711, −0.900382247523387602371549213780, −0.19621024229217829948481741662,
0.19621024229217829948481741662, 0.900382247523387602371549213780, 1.40528954866311429447308415711, 1.73610263099803189247974954658, 1.82041611211165762078027645365, 1.96727232681080610180130496397, 2.07608496722704828197574997567, 2.10943270936933276531839013596, 2.12174215673087104247097525244, 2.48009833454209043096434564647, 2.55484236272690200111100406747, 2.78618473632705026028466914791, 2.82304005995032521481622582497, 2.97194048793232155159349602487, 3.06203263807475550417502343270, 3.22249956800353534867804659320, 3.24691575059164648916771440846, 3.46866357297740168708394365089, 3.48502519311608542932796251612, 3.50207224093443734710015805363, 3.54179762518627065481637119395, 3.82837918064377410292606390095, 4.08253443869356656093441159910, 4.11693125411179153206936073441, 4.26023846643999984491931086043
Plot not available for L-functions of degree greater than 10.
