| L(s) = 1 | + 2·3-s + 9-s − 11-s + 2·13-s − 6·17-s − 4·19-s − 2·23-s − 4·27-s − 10·29-s + 8·31-s − 2·33-s − 8·37-s + 4·39-s − 2·41-s + 2·47-s − 7·49-s − 12·51-s − 8·57-s + 12·59-s − 10·61-s − 6·67-s − 4·69-s − 6·73-s − 12·79-s − 11·81-s + 16·83-s − 20·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.417·23-s − 0.769·27-s − 1.85·29-s + 1.43·31-s − 0.348·33-s − 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.291·47-s − 49-s − 1.68·51-s − 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.733·67-s − 0.481·69-s − 0.702·73-s − 1.35·79-s − 1.22·81-s + 1.75·83-s − 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−8.137296030013542999760814973380, −7.44768565041909769092975201090, −6.58547960716893830705713223683, −5.91622166242744518172165511635, −4.85769740309940293341350504866, −4.03622799462293727086828674780, −3.35384730409316744338498807656, −2.39205859293039716070712786825, −1.78135322666156763674745180325, 0,
1.78135322666156763674745180325, 2.39205859293039716070712786825, 3.35384730409316744338498807656, 4.03622799462293727086828674780, 4.85769740309940293341350504866, 5.91622166242744518172165511635, 6.58547960716893830705713223683, 7.44768565041909769092975201090, 8.137296030013542999760814973380
