| L(s) = 1 | + 7-s − 3·11-s − 4·13-s + 6·17-s − 4·19-s − 5·25-s + 6·29-s + 2·31-s − 7·37-s − 12·41-s − 7·43-s − 6·47-s + 49-s + 3·53-s − 6·59-s + 2·61-s − 13·67-s + 9·71-s + 8·73-s − 3·77-s − 79-s + 12·89-s − 4·91-s − 4·97-s − 12·101-s − 4·103-s + 15·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s − 25-s + 1.11·29-s + 0.359·31-s − 1.15·37-s − 1.87·41-s − 1.06·43-s − 0.875·47-s + 1/7·49-s + 0.412·53-s − 0.781·59-s + 0.256·61-s − 1.58·67-s + 1.06·71-s + 0.936·73-s − 0.341·77-s − 0.112·79-s + 1.27·89-s − 0.419·91-s − 0.406·97-s − 1.19·101-s − 0.394·103-s + 1.45·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−8.367405229597541969143450600182, −8.013715186180625594119400228016, −7.16969185229083907061004830786, −6.31891530104611065760601551687, −5.23813287136191298515733096075, −4.87092313996227080026581484887, −3.64041872315825866632821982347, −2.69339082526379787317114374377, −1.66109110854626091794561079002, 0,
1.66109110854626091794561079002, 2.69339082526379787317114374377, 3.64041872315825866632821982347, 4.87092313996227080026581484887, 5.23813287136191298515733096075, 6.31891530104611065760601551687, 7.16969185229083907061004830786, 8.013715186180625594119400228016, 8.367405229597541969143450600182
