| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 2·11-s − 2·13-s − 2·14-s + 16-s − 4·17-s − 2·19-s + 20-s + 2·22-s − 2·23-s + 25-s + 2·26-s + 2·28-s + 6·29-s + 8·31-s − 32-s + 4·34-s + 2·35-s + 2·37-s + 2·38-s − 40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.223·20-s + 0.426·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.685·34-s + 0.338·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.305718745210211973407229974090, −7.54556206549846058091530866302, −6.62836920415408922992208061668, −6.14193760908946642016010633870, −4.93295975188388434996561860222, −4.59975890079857617820027485194, −3.11843351995006321493189624931, −2.29334344669215674514520588732, −1.47279046569573052993068388951, 0,
1.47279046569573052993068388951, 2.29334344669215674514520588732, 3.11843351995006321493189624931, 4.59975890079857617820027485194, 4.93295975188388434996561860222, 6.14193760908946642016010633870, 6.62836920415408922992208061668, 7.54556206549846058091530866302, 8.305718745210211973407229974090
