| L(s) = 1 | + 2·3-s + 3·5-s + 7-s + 9-s + 6·15-s − 6·17-s − 7·19-s + 2·21-s − 3·23-s + 4·25-s − 4·27-s − 9·29-s + 5·31-s + 3·35-s − 2·37-s + 6·41-s + 43-s + 3·45-s + 3·47-s + 49-s − 12·51-s − 9·53-s − 14·57-s − 10·61-s + 63-s + 14·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.54·15-s − 1.45·17-s − 1.60·19-s + 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.769·27-s − 1.67·29-s + 0.898·31-s + 0.507·35-s − 0.328·37-s + 0.937·41-s + 0.152·43-s + 0.447·45-s + 0.437·47-s + 1/7·49-s − 1.68·51-s − 1.23·53-s − 1.85·57-s − 1.28·61-s + 0.125·63-s + 1.71·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−15.71163108622258, −15.42155316186646, −14.69663262643249, −14.41165625275906, −13.75712996210193, −13.48041833219154, −12.89119989933543, −12.46370022532874, −11.35463119866741, −11.03214488768436, −10.32294309257686, −9.723157226676027, −9.185081086236078, −8.758617607888968, −8.279569651913566, −7.568220735166279, −6.864923528347498, −6.026108670982932, −5.869798823807070, −4.732383264656515, −4.257521965264760, −3.430710537522067, −2.421197616216489, −2.194109652381734, −1.569670561912751, 0,
1.569670561912751, 2.194109652381734, 2.421197616216489, 3.430710537522067, 4.257521965264760, 4.732383264656515, 5.869798823807070, 6.026108670982932, 6.864923528347498, 7.568220735166279, 8.279569651913566, 8.758617607888968, 9.185081086236078, 9.723157226676027, 10.32294309257686, 11.03214488768436, 11.35463119866741, 12.46370022532874, 12.89119989933543, 13.48041833219154, 13.75712996210193, 14.41165625275906, 14.69663262643249, 15.42155316186646, 15.71163108622258
