| L(s) = 1 | + 2-s + 4-s − 2·5-s − 2·7-s + 8-s − 2·10-s − 2·11-s + 4·13-s − 2·14-s + 16-s + 4·17-s − 4·19-s − 2·20-s − 2·22-s + 6·23-s − 25-s + 4·26-s − 2·28-s + 4·31-s + 32-s + 4·34-s + 4·35-s − 37-s − 4·38-s − 2·40-s + 10·41-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 0.353·8-s − 0.632·10-s − 0.603·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.447·20-s − 0.426·22-s + 1.25·23-s − 1/5·25-s + 0.784·26-s − 0.377·28-s + 0.718·31-s + 0.176·32-s + 0.685·34-s + 0.676·35-s − 0.164·37-s − 0.648·38-s − 0.316·40-s + 1.56·41-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.49441150736627, −13.12684146133750, −12.86467449316966, −12.24554880514104, −11.89303283484933, −11.31825061388761, −10.84874765582704, −10.50231746981048, −9.926286621345777, −9.323931081522686, −8.683460858168734, −8.267413387864142, −7.767450193441343, −7.152988695335650, −6.842592187246268, −6.037396541680373, −5.810987588279873, −5.188119016371117, −4.398604723314504, −4.064283392220384, −3.545383110126830, −2.876870951835400, −2.630401501105585, −1.516211527237410, −0.8760594796931341, 0,
0.8760594796931341, 1.516211527237410, 2.630401501105585, 2.876870951835400, 3.545383110126830, 4.064283392220384, 4.398604723314504, 5.188119016371117, 5.810987588279873, 6.037396541680373, 6.842592187246268, 7.152988695335650, 7.767450193441343, 8.267413387864142, 8.683460858168734, 9.323931081522686, 9.926286621345777, 10.50231746981048, 10.84874765582704, 11.31825061388761, 11.89303283484933, 12.24554880514104, 12.86467449316966, 13.12684146133750, 13.49441150736627
