| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 6·11-s + 4·14-s + 16-s − 4·17-s − 2·19-s + 6·22-s − 6·23-s − 4·28-s + 10·29-s − 4·31-s − 32-s + 4·34-s − 6·37-s + 2·38-s + 10·41-s − 6·44-s + 6·46-s − 8·47-s + 9·49-s − 6·53-s + 4·56-s − 10·58-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.80·11-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 1.27·22-s − 1.25·23-s − 0.755·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.986·37-s + 0.324·38-s + 1.56·41-s − 0.904·44-s + 0.884·46-s − 1.16·47-s + 9/7·49-s − 0.824·53-s + 0.534·56-s − 1.31·58-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−14.28582554621035, −13.59315322474563, −13.31435576264192, −12.72821768305575, −12.38971303498985, −11.89618835148040, −11.04573920973280, −10.57987561335754, −10.35741231898311, −9.780146483997845, −9.313594210491254, −8.748807900911145, −8.202702685613629, −7.731320634207664, −7.176875776593321, −6.526142260465134, −6.162069371899253, −5.670561088659953, −4.807032794493331, −4.332654765215366, −3.363981182403311, −2.958776849379173, −2.383248287999147, −1.764354474593499, −0.5106769150144708, 0,
0.5106769150144708, 1.764354474593499, 2.383248287999147, 2.958776849379173, 3.363981182403311, 4.332654765215366, 4.807032794493331, 5.670561088659953, 6.162069371899253, 6.526142260465134, 7.176875776593321, 7.731320634207664, 8.202702685613629, 8.748807900911145, 9.313594210491254, 9.780146483997845, 10.35741231898311, 10.57987561335754, 11.04573920973280, 11.89618835148040, 12.38971303498985, 12.72821768305575, 13.31435576264192, 13.59315322474563, 14.28582554621035
