| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s − 2·11-s + 13-s + 15-s − 5·19-s − 3·21-s − 3·23-s − 4·25-s − 27-s − 4·29-s + 4·31-s + 2·33-s − 3·35-s + 2·37-s − 39-s − 2·41-s + 4·43-s − 45-s + 4·47-s + 2·49-s + 8·53-s + 2·55-s + 5·57-s + 14·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.258·15-s − 1.14·19-s − 0.654·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s − 0.742·29-s + 0.718·31-s + 0.348·33-s − 0.507·35-s + 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s + 2/7·49-s + 1.09·53-s + 0.269·55-s + 0.662·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.44445437860919, −12.83408673375911, −12.40810805960944, −11.84855148697126, −11.54153895759988, −11.07250444634825, −10.68322050616996, −10.19259301272106, −9.775538974583144, −8.972073010712937, −8.539029454513914, −8.107454991124812, −7.592451140199831, −7.321945749639788, −6.482925138717174, −6.094989351117523, −5.518659093697425, −5.061212714285123, −4.491702824270460, −4.019436591757588, −3.629663297802924, −2.566160643482284, −2.164061372001945, −1.509867897023443, −0.7332766406018260, 0,
0.7332766406018260, 1.509867897023443, 2.164061372001945, 2.566160643482284, 3.629663297802924, 4.019436591757588, 4.491702824270460, 5.061212714285123, 5.518659093697425, 6.094989351117523, 6.482925138717174, 7.321945749639788, 7.592451140199831, 8.107454991124812, 8.539029454513914, 8.972073010712937, 9.775538974583144, 10.19259301272106, 10.68322050616996, 11.07250444634825, 11.54153895759988, 11.84855148697126, 12.40810805960944, 12.83408673375911, 13.44445437860919
