| L(s) = 1 | + 7-s + 2·11-s + 2·13-s − 4·17-s − 4·19-s − 6·23-s − 5·25-s − 2·29-s + 6·37-s − 8·41-s − 8·43-s − 4·47-s + 49-s − 6·53-s + 14·61-s + 4·67-s − 2·71-s − 2·73-s + 2·77-s − 4·79-s − 12·83-s + 2·91-s + 6·97-s + 12·101-s + 8·103-s − 6·107-s + 18·109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.554·13-s − 0.970·17-s − 0.917·19-s − 1.25·23-s − 25-s − 0.371·29-s + 0.986·37-s − 1.24·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.824·53-s + 1.79·61-s + 0.488·67-s − 0.237·71-s − 0.234·73-s + 0.227·77-s − 0.450·79-s − 1.31·83-s + 0.209·91-s + 0.609·97-s + 1.19·101-s + 0.788·103-s − 0.580·107-s + 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.321087869606634401891245997354, −7.34793644413038459956712918109, −6.46623138891639762114972008358, −6.05094504092373992982078642379, −5.00596561703897150201379031400, −4.18067107542353056162359629825, −3.60845197465265311641819294253, −2.28613511580267472263897051908, −1.56841995500689962506292115998, 0,
1.56841995500689962506292115998, 2.28613511580267472263897051908, 3.60845197465265311641819294253, 4.18067107542353056162359629825, 5.00596561703897150201379031400, 6.05094504092373992982078642379, 6.46623138891639762114972008358, 7.34793644413038459956712918109, 8.321087869606634401891245997354
