| L(s) = 1 | + 67·9-s + 42·11-s − 204·19-s − 258·29-s + 640·31-s − 188·41-s − 98·49-s − 1.02e3·59-s − 452·61-s + 112·71-s − 2.59e3·79-s + 2.07e3·81-s − 1.09e3·89-s + 2.81e3·99-s − 280·101-s − 2.95e3·109-s − 4.18e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 311·169-s + ⋯ |
| L(s) = 1 | + 2.48·9-s + 1.15·11-s − 2.46·19-s − 1.65·29-s + 3.70·31-s − 0.716·41-s − 2/7·49-s − 2.25·59-s − 0.948·61-s + 0.187·71-s − 3.69·79-s + 2.84·81-s − 1.30·89-s + 2.85·99-s − 0.275·101-s − 2.59·109-s − 3.14·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.141·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01877620525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01877620525\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 67 T^{2} + 2416 T^{4} - 67 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 21 T + 2754 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 311 T^{2} + 7505592 T^{4} - 311 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2871 T^{2} + 39317744 T^{4} - 2871 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 102 T + 16246 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 32216 T^{2} + 535648110 T^{4} - 32216 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 129 T + 43284 T^{2} + 129 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 320 T + 80510 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 143244 T^{2} + 10204632502 T^{4} - 143244 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 94 T + 131218 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 273752 T^{2} + 30894105726 T^{4} - 273752 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 394243 T^{2} + 60414549464 T^{4} - 394243 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 512320 T^{2} + 108789156206 T^{4} - 512320 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 512 T + 247366 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 226 T + 428114 T^{2} + 226 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 745164 T^{2} + 270203734870 T^{4} - 745164 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 56 T + 659374 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 295356 T^{2} - 17377275290 T^{4} - 295356 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1299 T + 1390390 T^{2} + 1299 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 913188 T^{2} + 587868432374 T^{4} + 913188 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 546 T + 343842 T^{2} + 546 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 781681 T^{2} + 196993826592 T^{4} + 781681 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.47013684407008706475005977127, −6.32008658533302122538072383396, −6.17206775214756408508309886884, −5.83504964658209104933924099292, −5.61267731492464935385889392117, −5.37812298790810630577350332709, −5.00607089242555600940266183771, −4.57647095552185499671406165616, −4.48594508832571429582032372344, −4.48577097742272058482215825701, −4.35127727092287869474669748563, −4.02921944309691121312384883630, −3.82121194402024102143385941800, −3.52702625248941281442973361374, −3.23612404491590756933041070199, −2.90361289945450604899387107729, −2.62691020933837482333606479424, −2.31971294062190783430694814386, −2.11789391513811598092936206407, −1.59384793561737100311554564604, −1.44923130298634689824617534285, −1.26506110830427708390383609014, −1.20067917861148291472594361753, −0.50492939321116025110337415184, −0.01487574382765514623396181328,
0.01487574382765514623396181328, 0.50492939321116025110337415184, 1.20067917861148291472594361753, 1.26506110830427708390383609014, 1.44923130298634689824617534285, 1.59384793561737100311554564604, 2.11789391513811598092936206407, 2.31971294062190783430694814386, 2.62691020933837482333606479424, 2.90361289945450604899387107729, 3.23612404491590756933041070199, 3.52702625248941281442973361374, 3.82121194402024102143385941800, 4.02921944309691121312384883630, 4.35127727092287869474669748563, 4.48577097742272058482215825701, 4.48594508832571429582032372344, 4.57647095552185499671406165616, 5.00607089242555600940266183771, 5.37812298790810630577350332709, 5.61267731492464935385889392117, 5.83504964658209104933924099292, 6.17206775214756408508309886884, 6.32008658533302122538072383396, 6.47013684407008706475005977127
