| L(s) = 1 | − 3-s + 9-s − 4·11-s + 13-s + 7·17-s − 23-s − 5·25-s − 27-s + 6·29-s − 31-s + 4·33-s + 7·37-s − 39-s + 5·41-s − 11·43-s + 10·47-s − 7·49-s − 7·51-s + 11·59-s − 61-s − 13·67-s + 69-s + 8·71-s + 6·73-s + 5·75-s − 14·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.69·17-s − 0.208·23-s − 25-s − 0.192·27-s + 1.11·29-s − 0.179·31-s + 0.696·33-s + 1.15·37-s − 0.160·39-s + 0.780·41-s − 1.67·43-s + 1.45·47-s − 49-s − 0.980·51-s + 1.43·59-s − 0.128·61-s − 1.58·67-s + 0.120·69-s + 0.949·71-s + 0.702·73-s + 0.577·75-s − 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758210277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758210277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.59201313061185, −13.01888652629211, −12.74731426465228, −12.03705957608973, −11.80211801316430, −11.24698842834675, −10.61245937987104, −10.23514091590367, −9.883422298478806, −9.371169688975739, −8.585698385553872, −8.040247722685104, −7.711555813615004, −7.271690195817756, −6.407145116472636, −6.104448594689916, −5.361181952394867, −5.243056845964529, −4.438896581794829, −3.864514765460534, −3.192768475165077, −2.645621214348878, −1.903627474496162, −1.105489764441038, −0.4761766377570125,
0.4761766377570125, 1.105489764441038, 1.903627474496162, 2.645621214348878, 3.192768475165077, 3.864514765460534, 4.438896581794829, 5.243056845964529, 5.361181952394867, 6.104448594689916, 6.407145116472636, 7.271690195817756, 7.711555813615004, 8.040247722685104, 8.585698385553872, 9.371169688975739, 9.883422298478806, 10.23514091590367, 10.61245937987104, 11.24698842834675, 11.80211801316430, 12.03705957608973, 12.74731426465228, 13.01888652629211, 13.59201313061185
