| L(s) = 1 | + 2·5-s − 2·17-s − 4·19-s − 25-s − 6·29-s + 2·37-s + 6·41-s + 12·43-s + 4·47-s − 7·49-s − 6·53-s + 8·59-s − 2·61-s + 4·67-s + 12·71-s + 14·73-s − 8·83-s − 4·85-s − 18·89-s − 8·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.328·37-s + 0.937·41-s + 1.82·43-s + 0.583·47-s − 49-s − 0.824·53-s + 1.04·59-s − 0.256·61-s + 0.488·67-s + 1.42·71-s + 1.63·73-s − 0.878·83-s − 0.433·85-s − 1.90·89-s − 0.820·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.277261627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.277261627\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−15.43827730585471, −14.82060728035307, −14.19126924842522, −13.93892363842929, −13.15800128560726, −12.78148637864056, −12.38987320429495, −11.43129412802085, −11.01740008959629, −10.59497351897616, −9.729903271119231, −9.467994458358749, −8.914369148714350, −8.140810097512953, −7.678562928529123, −6.805106789235408, −6.415332893502476, −5.658213114967349, −5.343746697602670, −4.288989831227480, −3.981204015677079, −2.887865704275786, −2.248597385152366, −1.678675378301531, −0.5862741720591160,
0.5862741720591160, 1.678675378301531, 2.248597385152366, 2.887865704275786, 3.981204015677079, 4.288989831227480, 5.343746697602670, 5.658213114967349, 6.415332893502476, 6.805106789235408, 7.678562928529123, 8.140810097512953, 8.914369148714350, 9.467994458358749, 9.729903271119231, 10.59497351897616, 11.01740008959629, 11.43129412802085, 12.38987320429495, 12.78148637864056, 13.15800128560726, 13.93892363842929, 14.19126924842522, 14.82060728035307, 15.43827730585471
