| L(s) = 1 | − 2·5-s − 6·11-s − 6·13-s + 8·19-s + 4·23-s − 25-s + 8·31-s − 2·37-s − 12·43-s − 4·47-s − 7·49-s − 6·53-s + 12·55-s + 6·59-s + 14·61-s + 12·65-s − 2·73-s − 16·79-s − 2·83-s + 12·89-s − 16·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.80·11-s − 1.66·13-s + 1.83·19-s + 0.834·23-s − 1/5·25-s + 1.43·31-s − 0.328·37-s − 1.82·43-s − 0.583·47-s − 49-s − 0.824·53-s + 1.61·55-s + 0.781·59-s + 1.79·61-s + 1.48·65-s − 0.234·73-s − 1.80·79-s − 0.219·83-s + 1.27·89-s − 1.64·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6260490902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6260490902\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 1129 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 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 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.10580370110724, −12.93171862401459, −12.19108558079569, −11.81776076366159, −11.48322228085055, −11.02287021157637, −10.16473266140105, −9.968900390835001, −9.722280896669629, −8.851338587413714, −8.232227698295821, −7.930212813213911, −7.501906846191097, −7.076324123204092, −6.617296906712795, −5.620347656400613, −5.297381608884688, −4.716323750539348, −4.597694585230368, −3.353707935060704, −3.251853973582599, −2.600659766996702, −2.012300690260931, −1.022993899102844, −0.2586862116640704,
0.2586862116640704, 1.022993899102844, 2.012300690260931, 2.600659766996702, 3.251853973582599, 3.353707935060704, 4.597694585230368, 4.716323750539348, 5.297381608884688, 5.620347656400613, 6.617296906712795, 7.076324123204092, 7.501906846191097, 7.930212813213911, 8.232227698295821, 8.851338587413714, 9.722280896669629, 9.968900390835001, 10.16473266140105, 11.02287021157637, 11.48322228085055, 11.81776076366159, 12.19108558079569, 12.93171862401459, 13.10580370110724
