| L(s) = 1 | − 8·2-s + 6·3-s − 148·4-s + 470·5-s − 48·6-s + 1.85e3·8-s − 2.18e3·9-s − 3.76e3·10-s + 2.66e3·11-s − 888·12-s − 344·13-s + 2.82e3·15-s + 8.33e3·16-s + 8.46e3·17-s + 1.74e4·18-s + 3.52e4·19-s − 6.95e4·20-s − 2.12e4·22-s − 6.14e4·23-s + 1.11e4·24-s + 3.34e4·25-s + 2.75e3·26-s − 1.33e4·27-s + 1.79e5·29-s − 2.25e4·30-s + 5.71e4·31-s − 1.77e5·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.128·3-s − 1.15·4-s + 1.68·5-s − 0.0907·6-s + 1.28·8-s − 9-s − 1.18·10-s + 0.603·11-s − 0.148·12-s − 0.0434·13-s + 0.215·15-s + 0.508·16-s + 0.418·17-s + 0.707·18-s + 1.18·19-s − 1.94·20-s − 0.426·22-s − 1.05·23-s + 0.164·24-s + 0.427·25-s + 0.0307·26-s − 0.130·27-s + 1.36·29-s − 0.152·30-s + 0.344·31-s − 0.957·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.738300327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.738300327\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p^{3} T + 53 p^{2} T^{2} + p^{10} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 p T + 247 p^{2} T^{2} - 2 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 94 p T + 7499 p^{2} T^{2} - 94 p^{8} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 344 T + 109427178 T^{2} + 344 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8468 T + 1895926 p T^{2} - 8468 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 35280 T + 1565373638 T^{2} - 35280 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 61486 T + 6650536943 T^{2} + 61486 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 179040 T + 34622681578 T^{2} - 179040 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 57166 T + 31335570111 T^{2} - 57166 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 877698 T + 381609348827 T^{2} + 877698 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 283616 T + 125033442626 T^{2} - 283616 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 275484 T + 552841024778 T^{2} - 275484 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1662512 T + 1690275368222 T^{2} + 1662512 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1616484 T + 2738235417598 T^{2} - 1616484 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2454130 T + 5185075198823 T^{2} - 2454130 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6019176 T + 15292993001786 T^{2} - 6019176 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 174698 T + 2881438572447 T^{2} + 174698 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1151466 T + 6215147678071 T^{2} + 1151466 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 885944 T + 20004042345618 T^{2} + 885944 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3801460 T + 35726075376258 T^{2} - 3801460 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2282916 T + 53736100067578 T^{2} - 2282916 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13481970 T + 131461606905283 T^{2} - 13481970 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 68078 T + 159761571363987 T^{2} - 68078 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.788536519765423266431069588546, −9.667873454025425281532368433400, −8.859046955737133193034974604601, −8.828237609751519089973240391609, −8.259676396281221210547482420216, −8.047795460706898228509215458453, −7.22391136379552327495543396142, −6.74403417378233929911400520969, −6.22836057370081050380151404294, −5.68673587725560420141671102358, −5.25432328943705442087172969189, −5.13954196019171425338311162307, −4.29683881860688404386381021681, −3.67517481230392778620551500790, −3.29102974102346022406622071841, −2.46473724347136917457290077604, −1.97826826757972134575180937062, −1.47753333455928911402068860968, −0.70329710436939530993639656326, −0.52563726521101199961116635039,
0.52563726521101199961116635039, 0.70329710436939530993639656326, 1.47753333455928911402068860968, 1.97826826757972134575180937062, 2.46473724347136917457290077604, 3.29102974102346022406622071841, 3.67517481230392778620551500790, 4.29683881860688404386381021681, 5.13954196019171425338311162307, 5.25432328943705442087172969189, 5.68673587725560420141671102358, 6.22836057370081050380151404294, 6.74403417378233929911400520969, 7.22391136379552327495543396142, 8.047795460706898228509215458453, 8.259676396281221210547482420216, 8.828237609751519089973240391609, 8.859046955737133193034974604601, 9.667873454025425281532368433400, 9.788536519765423266431069588546
