| L(s) = 1 | − 2-s − 4-s + 5-s + 4·7-s + 3·8-s − 10-s − 4·11-s + 6·13-s − 4·14-s − 16-s + 2·17-s − 4·19-s − 20-s + 4·22-s + 23-s + 25-s − 6·26-s − 4·28-s + 10·29-s − 8·31-s − 5·32-s − 2·34-s + 4·35-s + 2·37-s + 4·38-s + 3·40-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.51·7-s + 1.06·8-s − 0.316·10-s − 1.20·11-s + 1.66·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 0.755·28-s + 1.85·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s + 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.474·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255276682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255276682\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−9.975144384021979445982805046052, −8.892039182635089349759381766028, −8.251418780433660953118526367100, −7.952706742196712345967880335530, −6.66130903538073920546111789283, −5.42325384197416427402124383567, −4.87713035050549148684233270206, −3.75783062785550806429257633674, −2.11230454009632946612683890961, −1.05140909064688750602011007914,
1.05140909064688750602011007914, 2.11230454009632946612683890961, 3.75783062785550806429257633674, 4.87713035050549148684233270206, 5.42325384197416427402124383567, 6.66130903538073920546111789283, 7.952706742196712345967880335530, 8.251418780433660953118526367100, 8.892039182635089349759381766028, 9.975144384021979445982805046052
