| L(s) = 1 | + 2·5-s − 7-s − 3·11-s + 2·17-s − 4·19-s − 25-s + 9·29-s − 5·31-s − 2·35-s + 4·37-s − 8·41-s − 6·43-s − 12·47-s − 6·49-s + 9·53-s − 6·55-s + 9·59-s − 8·67-s − 2·71-s + 5·73-s + 3·77-s − 11·79-s + 3·83-s + 4·85-s + 14·89-s − 8·95-s + 7·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.904·11-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 1.67·29-s − 0.898·31-s − 0.338·35-s + 0.657·37-s − 1.24·41-s − 0.914·43-s − 1.75·47-s − 6/7·49-s + 1.23·53-s − 0.809·55-s + 1.17·59-s − 0.977·67-s − 0.237·71-s + 0.585·73-s + 0.341·77-s − 1.23·79-s + 0.329·83-s + 0.433·85-s + 1.48·89-s − 0.820·95-s + 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517944322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517944322\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.25368391528731, −13.01854958610772, −12.52710688006373, −11.77625569836800, −11.60118047330475, −10.74920152642427, −10.31280250075592, −10.06890023180943, −9.636229744486831, −9.005536794236179, −8.397615224866779, −8.163717083379661, −7.455221203603063, −6.864529473262960, −6.346169282136914, −6.041782363895371, −5.287571147047708, −5.013130210922553, −4.374974876962977, −3.573386677994407, −3.111369113712425, −2.452023692534442, −1.956501897744911, −1.303520265638620, −0.3523694375045079,
0.3523694375045079, 1.303520265638620, 1.956501897744911, 2.452023692534442, 3.111369113712425, 3.573386677994407, 4.374974876962977, 5.013130210922553, 5.287571147047708, 6.041782363895371, 6.346169282136914, 6.864529473262960, 7.455221203603063, 8.163717083379661, 8.397615224866779, 9.005536794236179, 9.636229744486831, 10.06890023180943, 10.31280250075592, 10.74920152642427, 11.60118047330475, 11.77625569836800, 12.52710688006373, 13.01854958610772, 13.25368391528731
