| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 11-s − 2·13-s + 16-s − 5·17-s + 6·19-s + 20-s − 22-s + 7·23-s − 4·25-s − 2·26-s − 8·29-s − 10·31-s + 32-s − 5·34-s − 8·37-s + 6·38-s + 40-s − 7·41-s + 4·43-s − 44-s + 7·46-s + 47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.21·17-s + 1.37·19-s + 0.223·20-s − 0.213·22-s + 1.45·23-s − 4/5·25-s − 0.392·26-s − 1.48·29-s − 1.79·31-s + 0.176·32-s − 0.857·34-s − 1.31·37-s + 0.973·38-s + 0.158·40-s − 1.09·41-s + 0.609·43-s − 0.150·44-s + 1.03·46-s + 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−7.24121111862151547485485557402, −6.73249619027605054269766241166, −5.64649490074406581855629325796, −5.40886992437478628500847953046, −4.68197152220627813274676064118, −3.76661953295505867622592370184, −3.11791366099164814175505442744, −2.22104530024454952890704374539, −1.53545398176045998258155242534, 0,
1.53545398176045998258155242534, 2.22104530024454952890704374539, 3.11791366099164814175505442744, 3.76661953295505867622592370184, 4.68197152220627813274676064118, 5.40886992437478628500847953046, 5.64649490074406581855629325796, 6.73249619027605054269766241166, 7.24121111862151547485485557402
