| L(s) = 1 | − 3-s + 5-s + 9-s − 11-s + 13-s − 15-s + 4·17-s − 2·19-s + 23-s − 4·25-s − 27-s − 9·29-s − 4·31-s + 33-s − 6·37-s − 39-s − 41-s − 11·43-s + 45-s − 4·51-s − 10·53-s − 55-s + 2·57-s − 3·59-s − 5·61-s + 65-s − 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.258·15-s + 0.970·17-s − 0.458·19-s + 0.208·23-s − 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.718·31-s + 0.174·33-s − 0.986·37-s − 0.160·39-s − 0.156·41-s − 1.67·43-s + 0.149·45-s − 0.560·51-s − 1.37·53-s − 0.134·55-s + 0.264·57-s − 0.390·59-s − 0.640·61-s + 0.124·65-s − 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.12476345879659, −12.58244479108629, −12.14918118169779, −11.52560701836223, −11.37921422463793, −10.68666244874148, −10.36433982192667, −9.912107654577803, −9.495755695651256, −8.956973184064785, −8.513617032429632, −7.890069448633677, −7.423725060322565, −7.119759241111855, −6.353745302607366, −6.029665515929747, −5.539038439542687, −5.200477421077042, −4.600469139762625, −4.025664745507924, −3.367298908674989, −3.105524037543840, −2.114289653406457, −1.675680878735527, −1.266942618146186, 0, 0,
1.266942618146186, 1.675680878735527, 2.114289653406457, 3.105524037543840, 3.367298908674989, 4.025664745507924, 4.600469139762625, 5.200477421077042, 5.539038439542687, 6.029665515929747, 6.353745302607366, 7.119759241111855, 7.423725060322565, 7.890069448633677, 8.513617032429632, 8.956973184064785, 9.495755695651256, 9.912107654577803, 10.36433982192667, 10.68666244874148, 11.37921422463793, 11.52560701836223, 12.14918118169779, 12.58244479108629, 13.12476345879659
