| L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 2·11-s − 13-s + 2·17-s + 3·21-s + 6·23-s + 5·27-s + 10·29-s + 2·31-s + 2·33-s − 2·37-s + 39-s + 2·41-s + 4·43-s − 3·47-s + 2·49-s − 2·51-s + 4·53-s − 5·59-s − 12·61-s + 6·63-s + 8·67-s − 6·69-s − 13·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s + 0.485·17-s + 0.654·21-s + 1.25·23-s + 0.962·27-s + 1.85·29-s + 0.359·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.437·47-s + 2/7·49-s − 0.280·51-s + 0.549·53-s − 0.650·59-s − 1.53·61-s + 0.755·63-s + 0.977·67-s − 0.722·69-s − 1.54·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.55799270298361, −13.39393370408371, −12.58416915875017, −12.30620077184460, −12.04724477880239, −11.20299378014238, −10.93091685977671, −10.35137508509604, −9.981349727138802, −9.392601632119694, −8.969587519713649, −8.271803126819995, −7.993957496208503, −7.057220807248106, −6.852454516895850, −6.258577769973982, −5.713356035205257, −5.336024749296688, −4.681940721464962, −4.208668346781701, −3.132688877298660, −3.021889710395373, −2.517392380556933, −1.388036642620924, −0.6862850783593002, 0,
0.6862850783593002, 1.388036642620924, 2.517392380556933, 3.021889710395373, 3.132688877298660, 4.208668346781701, 4.681940721464962, 5.336024749296688, 5.713356035205257, 6.258577769973982, 6.852454516895850, 7.057220807248106, 7.993957496208503, 8.271803126819995, 8.969587519713649, 9.392601632119694, 9.981349727138802, 10.35137508509604, 10.93091685977671, 11.20299378014238, 12.04724477880239, 12.30620077184460, 12.58416915875017, 13.39393370408371, 13.55799270298361
