| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s + 2·13-s − 15-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s + 10·41-s + 45-s − 7·49-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s − 4·59-s − 10·61-s + 2·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.906903906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906903906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 223 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.84849723145572, −12.51066704496731, −12.18249027009395, −11.40860205002331, −11.22493829352193, −10.74027975220957, −10.27240232491358, −9.631834440504138, −9.313756549514622, −8.854975220133279, −8.269349584203098, −7.838753792128240, −7.059668471169069, −6.682018527992835, −6.220377877209225, −5.894552695464914, −5.348563646353662, −4.605690923337853, −4.081138658598222, −3.923774206936277, −2.956939359579421, −2.384937234373020, −1.648602313060984, −1.257181995698543, −0.4116417813422173,
0.4116417813422173, 1.257181995698543, 1.648602313060984, 2.384937234373020, 2.956939359579421, 3.923774206936277, 4.081138658598222, 4.605690923337853, 5.348563646353662, 5.894552695464914, 6.220377877209225, 6.682018527992835, 7.059668471169069, 7.838753792128240, 8.269349584203098, 8.854975220133279, 9.313756549514622, 9.631834440504138, 10.27240232491358, 10.74027975220957, 11.22493829352193, 11.40860205002331, 12.18249027009395, 12.51066704496731, 12.84849723145572
