| L(s) = 1 | − 3-s + 4·5-s + 4·7-s + 9-s + 11-s + 4·13-s − 4·15-s − 5·17-s − 5·19-s − 4·21-s + 2·23-s + 11·25-s − 27-s + 8·29-s − 4·31-s − 33-s + 16·35-s + 2·37-s − 4·39-s + 3·41-s + 4·45-s − 4·47-s + 9·49-s + 5·51-s − 12·53-s + 4·55-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 1.03·15-s − 1.21·17-s − 1.14·19-s − 0.872·21-s + 0.417·23-s + 11/5·25-s − 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.174·33-s + 2.70·35-s + 0.328·37-s − 0.640·39-s + 0.468·41-s + 0.596·45-s − 0.583·47-s + 9/7·49-s + 0.700·51-s − 1.64·53-s + 0.539·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−12.73297402151704, −12.54208285249506, −11.63798244679363, −11.19596700926899, −11.09143294304026, −10.46825464417707, −10.28524190286529, −9.564722432958790, −9.057769979868493, −8.732093501249808, −8.341791446174814, −7.788514729072183, −6.987254255685014, −6.616784637488908, −6.154073757443525, −5.929214635506668, −5.199525832544352, −4.888251590003112, −4.395256735346636, −3.969864410045307, −2.951560675608125, −2.452398455082927, −1.858456789395725, −1.414166056863015, −1.125029718032157, 0,
1.125029718032157, 1.414166056863015, 1.858456789395725, 2.452398455082927, 2.951560675608125, 3.969864410045307, 4.395256735346636, 4.888251590003112, 5.199525832544352, 5.929214635506668, 6.154073757443525, 6.616784637488908, 6.987254255685014, 7.788514729072183, 8.341791446174814, 8.732093501249808, 9.057769979868493, 9.564722432958790, 10.28524190286529, 10.46825464417707, 11.09143294304026, 11.19596700926899, 11.63798244679363, 12.54208285249506, 12.73297402151704
