| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s + 4·11-s + 13-s − 14-s + 16-s − 2·17-s − 3·18-s − 4·19-s + 4·22-s − 8·23-s + 26-s − 28-s − 2·29-s + 4·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s − 4·38-s − 6·41-s + 4·44-s − 8·46-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s + 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.917·19-s + 0.852·22-s − 1.66·23-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 − 1/2·36-s − 1.64·37-s − 0.648·38-s − 0.937·41-s + 0.603·44-s − 1.17·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 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.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 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.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 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.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.144982638821765981876457323803, −6.83315548318213432660205909235, −6.51238220126295394483260801605, −5.82474633559886749958467018457, −5.02524487839808832244639209570, −3.99584432984049190862842180737, −3.59650304472002591854313626644, −2.51727608734179409940874956555, −1.64707732856571780212133836156, 0,
1.64707732856571780212133836156, 2.51727608734179409940874956555, 3.59650304472002591854313626644, 3.99584432984049190862842180737, 5.02524487839808832244639209570, 5.82474633559886749958467018457, 6.51238220126295394483260801605, 6.83315548318213432660205909235, 8.144982638821765981876457323803
