| L(s) = 1 | − 4·5-s − 3·7-s + 4·13-s + 8·17-s + 6·19-s + 6·23-s + 11·25-s + 3·29-s − 4·31-s + 12·35-s − 5·37-s − 8·43-s + 10·47-s + 2·49-s − 12·53-s + 59-s + 2·61-s − 16·65-s − 10·67-s − 5·71-s − 13·73-s + 6·79-s − 2·83-s − 32·85-s − 12·91-s − 24·95-s + 7·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.13·7-s + 1.10·13-s + 1.94·17-s + 1.37·19-s + 1.25·23-s + 11/5·25-s + 0.557·29-s − 0.718·31-s + 2.02·35-s − 0.821·37-s − 1.21·43-s + 1.45·47-s + 2/7·49-s − 1.64·53-s + 0.130·59-s + 0.256·61-s − 1.98·65-s − 1.22·67-s − 0.593·71-s − 1.52·73-s + 0.675·79-s − 0.219·83-s − 3.47·85-s − 1.25·91-s − 2.46·95-s + 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.491254819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491254819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84323166786942, −12.29611666048900, −12.12643634260268, −11.57089970999150, −11.19624847417104, −10.65502757725599, −10.16060153134565, −9.725914947461890, −9.031944682855361, −8.755105701352956, −8.173325256888356, −7.636280558886101, −7.287716128588666, −6.965383889005358, −6.214530717840917, −5.776546524306082, −5.123163110969809, −4.685864386932218, −3.879187240537116, −3.465729124691005, −3.178662988809792, −2.915618528027460, −1.506709245745746, −1.023079518981015, −0.4165671153798789,
0.4165671153798789, 1.023079518981015, 1.506709245745746, 2.915618528027460, 3.178662988809792, 3.465729124691005, 3.879187240537116, 4.685864386932218, 5.123163110969809, 5.776546524306082, 6.214530717840917, 6.965383889005358, 7.287716128588666, 7.636280558886101, 8.173325256888356, 8.755105701352956, 9.031944682855361, 9.725914947461890, 10.16060153134565, 10.65502757725599, 11.19624847417104, 11.57089970999150, 12.12643634260268, 12.29611666048900, 12.84323166786942
