| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s + 5·13-s + 16-s − 3·17-s + 18-s − 6·19-s − 22-s − 3·23-s − 24-s + 5·26-s − 27-s − 5·29-s − 5·31-s + 32-s + 33-s − 3·34-s + 36-s + 8·37-s − 6·38-s − 5·39-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 − 0.301·11-s − 0.288·12-s + 1.38·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.213·22-s − 0.625·23-s − 0.204·24-s + 0.980·26-s − 0.192·27-s − 0.928·29-s − 0.898·31-s + 0.176·32-s + 0.174·33-s − 0.514·34-s + 1/6·36-s + 1.31·37-s − 0.973·38-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.14847469999625, −13.66191422341753, −13.11981052702167, −12.76206931655983, −12.50732741824721, −11.63146165268758, −11.16513916921528, −10.98549123822700, −10.51973614178637, −9.770908530020177, −9.268080510821395, −8.583429529804740, −8.100913183280463, −7.577257118532164, −6.806149748262716, −6.363099531920118, −6.036983914794594, −5.428379808272727, −4.857641560642067, −4.125135197597811, −3.892391700110482, −3.173908582549039, −2.199077726471943, −1.890735224388391, −0.9141736522253180, 0,
0.9141736522253180, 1.890735224388391, 2.199077726471943, 3.173908582549039, 3.892391700110482, 4.125135197597811, 4.857641560642067, 5.428379808272727, 6.036983914794594, 6.363099531920118, 6.806149748262716, 7.577257118532164, 8.100913183280463, 8.583429529804740, 9.268080510821395, 9.770908530020177, 10.51973614178637, 10.98549123822700, 11.16513916921528, 11.63146165268758, 12.50732741824721, 12.76206931655983, 13.11981052702167, 13.66191422341753, 14.14847469999625
