| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 2·13-s + 16-s + 2·17-s + 4·19-s − 20-s + 8·23-s + 25-s + 2·26-s − 2·29-s − 4·31-s + 32-s + 2·34-s − 2·37-s + 4·38-s − 40-s − 10·41-s + 12·43-s + 8·46-s + 8·47-s + 50-s + 2·52-s + 53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s + 1.82·43-s + 1.17·46-s + 1.16·47-s + 0.141·50-s + 0.277·52-s + 0.137·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 53 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.04364513008961, −12.75850158598207, −12.24114826686408, −11.81551629683840, −11.34558939131184, −10.89374743322102, −10.60056810517331, −9.951743930015824, −9.330283043546985, −8.982323472885023, −8.415141292432706, −7.861969462808838, −7.291584476466276, −7.060495040754552, −6.515182243652724, −5.716305395393309, −5.488848125843854, −4.988513518039161, −4.351428271063431, −3.785752951770890, −3.409938651492987, −2.848359021909446, −2.306724669264797, −1.314142616508238, −1.067248393809381, 0,
1.067248393809381, 1.314142616508238, 2.306724669264797, 2.848359021909446, 3.409938651492987, 3.785752951770890, 4.351428271063431, 4.988513518039161, 5.488848125843854, 5.716305395393309, 6.515182243652724, 7.060495040754552, 7.291584476466276, 7.861969462808838, 8.415141292432706, 8.982323472885023, 9.330283043546985, 9.951743930015824, 10.60056810517331, 10.89374743322102, 11.34558939131184, 11.81551629683840, 12.24114826686408, 12.75850158598207, 13.04364513008961
