| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s − 5·13-s + 16-s − 17-s − 18-s − 2·23-s − 24-s + 5·26-s + 27-s − 7·29-s + 9·31-s − 32-s + 34-s + 36-s − 4·37-s − 5·39-s + 11·41-s + 8·43-s + 2·46-s − 11·47-s + 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.38·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.417·23-s − 0.204·24-s + 0.980·26-s + 0.192·27-s − 1.29·29-s + 1.61·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.657·37-s − 0.800·39-s + 1.71·41-s + 1.21·43-s + 0.294·46-s − 1.60·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.87670293772523, −13.33950279409424, −12.60018259006135, −12.36324954958507, −11.92334622358037, −11.17546928038067, −10.83459432301211, −10.30331807121978, −9.606236613157815, −9.464577311215273, −9.070652443182989, −8.240559366914519, −7.846969353195539, −7.618063215164614, −6.892734707311173, −6.480663083557877, −5.854531519831550, −5.169872141303329, −4.602140674345389, −4.061878955791075, −3.315595999275786, −2.715503401896758, −2.240267235649428, −1.669742685655179, −0.7909149136831292, 0,
0.7909149136831292, 1.669742685655179, 2.240267235649428, 2.715503401896758, 3.315595999275786, 4.061878955791075, 4.602140674345389, 5.169872141303329, 5.854531519831550, 6.480663083557877, 6.892734707311173, 7.618063215164614, 7.846969353195539, 8.240559366914519, 9.070652443182989, 9.464577311215273, 9.606236613157815, 10.30331807121978, 10.83459432301211, 11.17546928038067, 11.92334622358037, 12.36324954958507, 12.60018259006135, 13.33950279409424, 13.87670293772523
