| L(s) = 1 | − 2·5-s − 4·7-s − 4·11-s + 2·13-s − 2·17-s − 25-s − 2·29-s + 8·35-s − 10·37-s + 6·41-s − 8·43-s − 8·47-s + 9·49-s + 6·53-s + 8·55-s + 4·59-s + 14·61-s − 4·65-s − 8·67-s − 8·71-s − 6·73-s + 16·77-s + 12·79-s − 12·83-s + 4·85-s − 2·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 1/5·25-s − 0.371·29-s + 1.35·35-s − 1.64·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + 1.79·61-s − 0.496·65-s − 0.977·67-s − 0.949·71-s − 0.702·73-s + 1.82·77-s + 1.35·79-s − 1.31·83-s + 0.433·85-s − 0.211·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 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
−13.22915988909760, −12.66118221301471, −12.33191469391462, −11.72060175972178, −11.33347096178323, −10.89447841149990, −10.23215213392039, −10.05180452940339, −9.583925974408825, −8.776125098545530, −8.622287031449482, −8.108514794993621, −7.344933680448641, −7.254353345518781, −6.604353173170779, −6.132774695302897, −5.619188428760750, −5.117435534540341, −4.483527660398645, −3.913593674131372, −3.386812457323128, −3.170756191750209, −2.421469703213503, −1.875471695964714, −0.8964397036053468, 0, 0,
0.8964397036053468, 1.875471695964714, 2.421469703213503, 3.170756191750209, 3.386812457323128, 3.913593674131372, 4.483527660398645, 5.117435534540341, 5.619188428760750, 6.132774695302897, 6.604353173170779, 7.254353345518781, 7.344933680448641, 8.108514794993621, 8.622287031449482, 8.776125098545530, 9.583925974408825, 10.05180452940339, 10.23215213392039, 10.89447841149990, 11.33347096178323, 11.72060175972178, 12.33191469391462, 12.66118221301471, 13.22915988909760
