| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 9-s − 11-s − 13-s + 2·15-s − 2·17-s − 8·21-s − 4·23-s + 25-s + 4·27-s + 4·29-s − 4·31-s + 2·33-s − 4·35-s + 10·37-s + 2·39-s − 8·41-s + 8·43-s − 45-s − 2·47-s + 9·49-s + 4·51-s + 4·53-s + 55-s − 4·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s − 0.485·17-s − 1.74·21-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.742·29-s − 0.718·31-s + 0.348·33-s − 0.676·35-s + 1.64·37-s + 0.320·39-s − 1.24·41-s + 1.21·43-s − 0.149·45-s − 0.291·47-s + 9/7·49-s + 0.560·51-s + 0.549·53-s + 0.134·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 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.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−16.84961262904341, −16.18741965998450, −15.73472869406204, −14.95844466890444, −14.58211651980585, −13.96251174873164, −13.25150507087408, −12.51537575640098, −11.85986483650283, −11.66488310237016, −10.96970239383755, −10.65686402806857, −9.958283343271548, −9.002360897895038, −8.405722286621172, −7.782655152360313, −7.304279857754719, −6.402180457302359, −5.823132218734814, −5.126186002526123, −4.614418221345161, −4.094738431217433, −2.879644723868068, −1.979755587792820, −1.034710967249750, 0,
1.034710967249750, 1.979755587792820, 2.879644723868068, 4.094738431217433, 4.614418221345161, 5.126186002526123, 5.823132218734814, 6.402180457302359, 7.304279857754719, 7.782655152360313, 8.405722286621172, 9.002360897895038, 9.958283343271548, 10.65686402806857, 10.96970239383755, 11.66488310237016, 11.85986483650283, 12.51537575640098, 13.25150507087408, 13.96251174873164, 14.58211651980585, 14.95844466890444, 15.73472869406204, 16.18741965998450, 16.84961262904341
