| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 2·11-s − 15-s + 2·17-s + 2·19-s + 21-s + 4·23-s + 25-s + 27-s − 2·29-s + 6·31-s − 2·33-s − 35-s + 6·37-s − 6·41-s − 4·43-s − 45-s + 49-s + 2·51-s + 8·53-s + 2·55-s + 2·57-s − 10·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.348·33-s − 0.169·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 1.09·53-s + 0.269·55-s + 0.264·57-s − 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.046542722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.046542722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.98989187127348, −13.78804170459332, −13.02075121143027, −12.83885613062461, −12.02089855001185, −11.68908231687490, −11.16023724525394, −10.50880623297238, −10.15232132010340, −9.464807383355437, −9.091233624090761, −8.220963797100187, −8.148138814447292, −7.569308749500025, −6.923048613866554, −6.529847012666185, −5.500890286063240, −5.277022761106270, −4.480099931827398, −4.044904474247256, −3.186798846133897, −2.895245822229660, −2.103832957409757, −1.307244323041558, −0.5897013420760620,
0.5897013420760620, 1.307244323041558, 2.103832957409757, 2.895245822229660, 3.186798846133897, 4.044904474247256, 4.480099931827398, 5.277022761106270, 5.500890286063240, 6.529847012666185, 6.923048613866554, 7.569308749500025, 8.148138814447292, 8.220963797100187, 9.091233624090761, 9.464807383355437, 10.15232132010340, 10.50880623297238, 11.16023724525394, 11.68908231687490, 12.02089855001185, 12.83885613062461, 13.02075121143027, 13.78804170459332, 13.98989187127348
