| L(s) = 1 | − 3-s − 2·7-s + 9-s + 13-s + 4·17-s + 6·19-s + 2·21-s + 6·23-s − 27-s − 4·29-s − 8·31-s − 6·37-s − 39-s + 6·41-s − 4·43-s + 8·47-s − 3·49-s − 4·51-s + 2·53-s − 6·57-s + 2·61-s − 2·63-s + 4·67-s − 6·69-s + 8·71-s − 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.970·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.986·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s − 0.794·57-s + 0.256·61-s − 0.251·63-s + 0.488·67-s − 0.722·69-s + 0.949·71-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−14.58220157929277, −13.92393749231530, −13.49883473411296, −12.87367559604585, −12.53237391442216, −12.09324708091896, −11.34435909079757, −11.11142861160238, −10.48660354137172, −9.858564313792030, −9.508681006667130, −9.012853540210837, −8.368468715597079, −7.545300937878581, −7.223253143603943, −6.772133394112136, −5.992284876401992, −5.417757242189822, −5.289631435222758, −4.342421401968482, −3.540181580241126, −3.321974350199094, −2.484010667647517, −1.495988719860811, −0.9265120034138520, 0,
0.9265120034138520, 1.495988719860811, 2.484010667647517, 3.321974350199094, 3.540181580241126, 4.342421401968482, 5.289631435222758, 5.417757242189822, 5.992284876401992, 6.772133394112136, 7.223253143603943, 7.545300937878581, 8.368468715597079, 9.012853540210837, 9.508681006667130, 9.858564313792030, 10.48660354137172, 11.11142861160238, 11.34435909079757, 12.09324708091896, 12.53237391442216, 12.87367559604585, 13.49883473411296, 13.92393749231530, 14.58220157929277
