| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 5·13-s − 14-s + 16-s + 6·17-s − 23-s − 5·25-s + 5·26-s − 28-s − 6·29-s − 7·31-s + 32-s + 6·34-s + 5·37-s − 6·41-s − 7·43-s − 46-s + 12·47-s − 6·49-s − 5·50-s + 5·52-s − 6·53-s − 56-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.208·23-s − 25-s + 0.980·26-s − 0.188·28-s − 1.11·29-s − 1.25·31-s + 0.176·32-s + 1.02·34-s + 0.821·37-s − 0.937·41-s − 1.06·43-s − 0.147·46-s + 1.75·47-s − 6/7·49-s − 0.707·50-s + 0.693·52-s − 0.824·53-s − 0.133·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.780735063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.780735063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−13.39296917532810, −12.99116487622761, −12.41059706147726, −11.99885302039932, −11.44231477689494, −11.14057397994656, −10.47323886893549, −10.12978459406802, −9.486387478688839, −9.092316253099841, −8.440124615428649, −7.786188401646696, −7.598582742415009, −6.881666537897158, −6.220834585452367, −5.964427204647695, −5.414258500887472, −4.986245296400979, −4.036409130869881, −3.729423625815976, −3.362735163254754, −2.685593973464043, −1.790033050586273, −1.450239200082433, −0.4964937562303246,
0.4964937562303246, 1.450239200082433, 1.790033050586273, 2.685593973464043, 3.362735163254754, 3.729423625815976, 4.036409130869881, 4.986245296400979, 5.414258500887472, 5.964427204647695, 6.220834585452367, 6.881666537897158, 7.598582742415009, 7.786188401646696, 8.440124615428649, 9.092316253099841, 9.486387478688839, 10.12978459406802, 10.47323886893549, 11.14057397994656, 11.44231477689494, 11.99885302039932, 12.41059706147726, 12.99116487622761, 13.39296917532810
