| L(s) = 1 | + 3·5-s + 5·7-s − 5·11-s + 2·13-s − 7·17-s − 19-s + 4·25-s + 8·29-s − 4·31-s + 15·35-s − 37-s − 6·41-s + 11·43-s + 5·47-s + 18·49-s − 10·53-s − 15·55-s − 6·59-s + 13·61-s + 6·65-s − 2·67-s − 10·71-s − 11·73-s − 25·77-s − 14·79-s − 12·83-s − 21·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.88·7-s − 1.50·11-s + 0.554·13-s − 1.69·17-s − 0.229·19-s + 4/5·25-s + 1.48·29-s − 0.718·31-s + 2.53·35-s − 0.164·37-s − 0.937·41-s + 1.67·43-s + 0.729·47-s + 18/7·49-s − 1.37·53-s − 2.02·55-s − 0.781·59-s + 1.66·61-s + 0.744·65-s − 0.244·67-s − 1.18·71-s − 1.28·73-s − 2.84·77-s − 1.57·79-s − 1.31·83-s − 2.27·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.58810987580601, −14.21903753414921, −13.70916276595223, −13.37810038175991, −12.86585635250058, −12.28808943926335, −11.47843767231491, −11.05565408147215, −10.65749301151445, −10.30705767846098, −9.600118064823393, −8.860249664164103, −8.486316255757202, −8.146990535438159, −7.301977673792646, −6.919680832418551, −5.902197672504738, −5.765233951853289, −5.001280503303461, −4.617126455345978, −4.073928160891396, −2.692940626419446, −2.522729636968893, −1.718112752388468, −1.293696294112502, 0,
1.293696294112502, 1.718112752388468, 2.522729636968893, 2.692940626419446, 4.073928160891396, 4.617126455345978, 5.001280503303461, 5.765233951853289, 5.902197672504738, 6.919680832418551, 7.301977673792646, 8.146990535438159, 8.486316255757202, 8.860249664164103, 9.600118064823393, 10.30705767846098, 10.65749301151445, 11.05565408147215, 11.47843767231491, 12.28808943926335, 12.86585635250058, 13.37810038175991, 13.70916276595223, 14.21903753414921, 14.58810987580601
