| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 4·11-s − 13-s + 14-s + 16-s − 2·17-s − 4·19-s + 20-s − 4·22-s − 8·23-s + 25-s − 26-s + 28-s + 2·29-s + 4·31-s + 32-s − 2·34-s + 35-s + 10·37-s − 4·38-s + 40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.169·35-s + 1.64·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 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
−7.62804721674456759552084853115, −6.47231625830282368194125388192, −6.10569279307278453702672576056, −5.36818941203578845860141412090, −4.54456498639708041776749270628, −4.20250764886632416003395697426, −2.89457772170522130166690257095, −2.42893179723478132752768103898, −1.55300350591201351517527159773, 0,
1.55300350591201351517527159773, 2.42893179723478132752768103898, 2.89457772170522130166690257095, 4.20250764886632416003395697426, 4.54456498639708041776749270628, 5.36818941203578845860141412090, 6.10569279307278453702672576056, 6.47231625830282368194125388192, 7.62804721674456759552084853115
