| L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s + 11-s − 15-s − 7·17-s + 5·19-s + 3·21-s + 6·23-s + 25-s − 5·27-s + 5·29-s − 3·31-s + 33-s − 3·35-s − 3·37-s − 2·41-s − 4·43-s + 2·45-s − 2·47-s + 2·49-s − 7·51-s − 53-s − 55-s + 5·57-s − 10·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s + 0.301·11-s − 0.258·15-s − 1.69·17-s + 1.14·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s − 0.538·31-s + 0.174·33-s − 0.507·35-s − 0.493·37-s − 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.291·47-s + 2/7·49-s − 0.980·51-s − 0.137·53-s − 0.134·55-s + 0.662·57-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.69812280053648, −13.16275449294134, −12.76541517161359, −11.88860306675704, −11.68654903391224, −11.25785665900822, −10.88406032992078, −10.33522411759370, −9.585734368257112, −9.051156040717670, −8.742660002370754, −8.353568228243437, −7.799477732261829, −7.365576187990501, −6.747119603121854, −6.370283228929568, −5.398341575137246, −5.151370075210997, −4.538639470423839, −4.077005298494757, −3.244990056531430, −2.980615591689670, −2.177085683900258, −1.653488619744549, −0.9020931252459941, 0,
0.9020931252459941, 1.653488619744549, 2.177085683900258, 2.980615591689670, 3.244990056531430, 4.077005298494757, 4.538639470423839, 5.151370075210997, 5.398341575137246, 6.370283228929568, 6.747119603121854, 7.365576187990501, 7.799477732261829, 8.353568228243437, 8.742660002370754, 9.051156040717670, 9.585734368257112, 10.33522411759370, 10.88406032992078, 11.25785665900822, 11.68654903391224, 11.88860306675704, 12.76541517161359, 13.16275449294134, 13.69812280053648
