| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·13-s + 15-s − 4·19-s − 21-s − 4·23-s + 25-s + 27-s − 6·29-s − 35-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 8·47-s + 49-s + 14·53-s − 4·57-s − 4·59-s + 2·61-s − 63-s − 2·65-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.917·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.169·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 1.92·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s − 0.248·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.16119444526848, −12.69880420978789, −12.23691431445597, −11.86982002077653, −11.22588833442960, −10.66584197710409, −10.17020381069483, −9.966050284690209, −9.342397167306338, −8.941569350257971, −8.458809203284874, −8.054477113464887, −7.387533951644802, −6.940437376545251, −6.615978641096519, −5.851579831348255, −5.521217537902379, −4.937438321164920, −4.228065945985232, −3.846374104836617, −3.282950002786463, −2.633291434853299, −2.028161778665325, −1.798064997125343, −0.7574104770931613, 0,
0.7574104770931613, 1.798064997125343, 2.028161778665325, 2.633291434853299, 3.282950002786463, 3.846374104836617, 4.228065945985232, 4.937438321164920, 5.521217537902379, 5.851579831348255, 6.615978641096519, 6.940437376545251, 7.387533951644802, 8.054477113464887, 8.458809203284874, 8.941569350257971, 9.342397167306338, 9.966050284690209, 10.17020381069483, 10.66584197710409, 11.22588833442960, 11.86982002077653, 12.23691431445597, 12.69880420978789, 13.16119444526848
