| L(s) = 1 | − 3-s + 9-s + 2·11-s + 5·19-s + 8·23-s − 27-s + 6·29-s + 5·31-s − 2·33-s − 2·37-s − 12·41-s − 8·43-s − 7·49-s + 2·53-s − 5·57-s + 2·59-s + 7·61-s + 12·67-s − 8·69-s − 10·71-s + 7·73-s − 13·79-s + 81-s − 12·83-s − 6·87-s − 10·89-s − 5·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.14·19-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.898·31-s − 0.348·33-s − 0.328·37-s − 1.87·41-s − 1.21·43-s − 49-s + 0.274·53-s − 0.662·57-s + 0.260·59-s + 0.896·61-s + 1.46·67-s − 0.963·69-s − 1.18·71-s + 0.819·73-s − 1.46·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s − 1.05·89-s − 0.518·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.76599011550396, −14.21407718508601, −13.75320096458793, −13.13258852488982, −12.80544847134657, −11.93515731866367, −11.76142611671933, −11.32927613377911, −10.64567132410890, −10.00645668373893, −9.793344467006049, −9.005363985976467, −8.495475021219593, −8.004488786064399, −7.112783646244399, −6.786772547733451, −6.414876012965201, −5.465705846499915, −5.116038569351748, −4.614665935600086, −3.782377364192459, −3.184337398809504, −2.585206410077371, −1.411564396670750, −1.112986593892160, 0,
1.112986593892160, 1.411564396670750, 2.585206410077371, 3.184337398809504, 3.782377364192459, 4.614665935600086, 5.116038569351748, 5.465705846499915, 6.414876012965201, 6.786772547733451, 7.112783646244399, 8.004488786064399, 8.495475021219593, 9.005363985976467, 9.793344467006049, 10.00645668373893, 10.64567132410890, 11.32927613377911, 11.76142611671933, 11.93515731866367, 12.80544847134657, 13.13258852488982, 13.75320096458793, 14.21407718508601, 14.76599011550396
