| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 3·11-s + 2·13-s − 15-s − 5·19-s − 21-s − 9·23-s + 25-s − 27-s − 31-s − 3·33-s + 35-s + 8·37-s − 2·39-s − 11·43-s + 45-s + 6·47-s − 6·49-s − 9·53-s + 3·55-s + 5·57-s − 6·59-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 + 0.904·11-s + 0.554·13-s − 0.258·15-s − 1.14·19-s − 0.218·21-s − 1.87·23-s + 1/5·25-s − 0.192·27-s − 0.179·31-s − 0.522·33-s + 0.169·35-s + 1.31·37-s − 0.320·39-s − 1.67·43-s + 0.149·45-s + 0.875·47-s − 6/7·49-s − 1.23·53-s + 0.404·55-s + 0.662·57-s − 0.781·59-s − 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.61023220700089171590357573164, −6.49013802337263731192870412097, −6.27737996598791397595787420575, −5.62366701115997278650629655733, −4.55940159613077654972365791268, −4.19330117938825177351727503547, −3.18912859696098795082984999010, −1.96902273126241198885408503153, −1.41017143710842771412297459165, 0,
1.41017143710842771412297459165, 1.96902273126241198885408503153, 3.18912859696098795082984999010, 4.19330117938825177351727503547, 4.55940159613077654972365791268, 5.62366701115997278650629655733, 6.27737996598791397595787420575, 6.49013802337263731192870412097, 7.61023220700089171590357573164
