| L(s) = 1 | + 5-s − 2·7-s − 3·9-s + 4·11-s + 13-s − 2·17-s + 2·19-s − 4·25-s + 5·29-s + 6·31-s − 2·35-s − 10·37-s − 9·41-s − 10·43-s − 3·45-s + 12·47-s − 3·49-s + 5·53-s + 4·55-s − 6·59-s + 61-s + 6·63-s + 65-s − 8·67-s + 6·71-s − 9·73-s − 8·77-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s − 4/5·25-s + 0.928·29-s + 1.07·31-s − 0.338·35-s − 1.64·37-s − 1.40·41-s − 1.52·43-s − 0.447·45-s + 1.75·47-s − 3/7·49-s + 0.686·53-s + 0.539·55-s − 0.781·59-s + 0.128·61-s + 0.755·63-s + 0.124·65-s − 0.977·67-s + 0.712·71-s − 1.05·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−7.32295991052039221455713805174, −6.47441753096201261336767337625, −6.30702367093817068236023038133, −5.45430427555878429776665709457, −4.69088219349123329382292327029, −3.67923401519012950034030302020, −3.18506790510831789891479405534, −2.24153758187359637415709228687, −1.26375935595834277084743031812, 0,
1.26375935595834277084743031812, 2.24153758187359637415709228687, 3.18506790510831789891479405534, 3.67923401519012950034030302020, 4.69088219349123329382292327029, 5.45430427555878429776665709457, 6.30702367093817068236023038133, 6.47441753096201261336767337625, 7.32295991052039221455713805174
