| L(s) = 1 | − 2·5-s − 4·7-s + 4·13-s + 6·17-s + 23-s − 25-s + 8·31-s + 8·35-s − 2·37-s − 8·47-s + 9·49-s − 6·53-s + 4·59-s − 14·61-s − 8·65-s − 4·67-s − 8·71-s + 4·73-s + 4·83-s − 12·85-s − 2·89-s − 16·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.10·13-s + 1.45·17-s + 0.208·23-s − 1/5·25-s + 1.43·31-s + 1.35·35-s − 0.328·37-s − 1.16·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s − 1.79·61-s − 0.992·65-s − 0.488·67-s − 0.949·71-s + 0.468·73-s + 0.439·83-s − 1.30·85-s − 0.211·89-s − 1.67·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208238327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208238327\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.13416336088959, −12.53491893079004, −12.04419190738257, −11.85370677892953, −11.24159757638160, −10.62105874242569, −10.29116724652286, −9.709010508020548, −9.399382062246328, −8.775866915626847, −8.221589511528453, −7.876489866885307, −7.346023749244713, −6.765881039743419, −6.216215749291507, −6.017490990467293, −5.300125591041193, −4.628026334591121, −4.047505061027653, −3.484757084492611, −3.201771071706031, −2.765803605937540, −1.696724902257706, −1.045381550422634, −0.3580224646318706,
0.3580224646318706, 1.045381550422634, 1.696724902257706, 2.765803605937540, 3.201771071706031, 3.484757084492611, 4.047505061027653, 4.628026334591121, 5.300125591041193, 6.017490990467293, 6.216215749291507, 6.765881039743419, 7.346023749244713, 7.876489866885307, 8.221589511528453, 8.775866915626847, 9.399382062246328, 9.709010508020548, 10.29116724652286, 10.62105874242569, 11.24159757638160, 11.85370677892953, 12.04419190738257, 12.53491893079004, 13.13416336088959
