| L(s) = 1 | + 2·5-s − 7-s + 4·13-s + 19-s − 6·23-s − 25-s + 3·29-s + 9·31-s − 2·35-s − 6·37-s − 6·41-s + 6·43-s + 47-s + 49-s − 13·53-s − 7·59-s + 10·61-s + 8·65-s + 6·67-s + 6·71-s − 2·73-s − 16·79-s − 15·83-s + 14·89-s − 4·91-s + 2·95-s − 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.10·13-s + 0.229·19-s − 1.25·23-s − 1/5·25-s + 0.557·29-s + 1.61·31-s − 0.338·35-s − 0.986·37-s − 0.937·41-s + 0.914·43-s + 0.145·47-s + 1/7·49-s − 1.78·53-s − 0.911·59-s + 1.28·61-s + 0.992·65-s + 0.733·67-s + 0.712·71-s − 0.234·73-s − 1.80·79-s − 1.64·83-s + 1.48·89-s − 0.419·91-s + 0.205·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−14.04025386168388, −13.95482037600151, −13.44416234550850, −12.93307164460117, −12.38648201188819, −11.84643149324972, −11.41030260471910, −10.68226174294284, −10.29369626159000, −9.763375750729946, −9.478225287853162, −8.653360322703322, −8.347793163015824, −7.787574994873620, −6.980164409881849, −6.457466880518590, −6.054820943058662, −5.634312463634003, −4.917312554600385, −4.263532094558989, −3.643084276000016, −3.031055963425729, −2.350568635765130, −1.659425491928937, −1.054731066628609, 0,
1.054731066628609, 1.659425491928937, 2.350568635765130, 3.031055963425729, 3.643084276000016, 4.263532094558989, 4.917312554600385, 5.634312463634003, 6.054820943058662, 6.457466880518590, 6.980164409881849, 7.787574994873620, 8.347793163015824, 8.653360322703322, 9.478225287853162, 9.763375750729946, 10.29369626159000, 10.68226174294284, 11.41030260471910, 11.84643149324972, 12.38648201188819, 12.93307164460117, 13.44416234550850, 13.95482037600151, 14.04025386168388
