| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 6·17-s − 21-s − 23-s − 25-s − 27-s − 2·29-s − 4·31-s + 4·33-s − 2·35-s + 6·37-s − 2·39-s − 6·41-s − 12·43-s − 2·45-s + 12·47-s + 49-s − 6·51-s + 6·53-s + 8·55-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.218·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s − 0.338·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.67885422258497769779216024519, −6.95261553830085557770774646761, −5.95604988663979719795800768667, −5.42337022626503229617169061411, −4.80111153791336777277174054528, −3.86876834426994350817882843688, −3.32708992037196660019604706161, −2.19608361011235337150062851626, −1.07600596181236249452903536885, 0,
1.07600596181236249452903536885, 2.19608361011235337150062851626, 3.32708992037196660019604706161, 3.86876834426994350817882843688, 4.80111153791336777277174054528, 5.42337022626503229617169061411, 5.95604988663979719795800768667, 6.95261553830085557770774646761, 7.67885422258497769779216024519
