| L(s) = 1 | + 2·5-s − 11-s − 2·17-s − 6·19-s − 6·23-s − 25-s − 2·29-s − 2·37-s + 2·41-s − 6·43-s − 6·47-s − 7·49-s − 2·53-s − 2·55-s + 8·61-s − 12·67-s + 6·71-s + 6·73-s − 12·79-s + 12·83-s − 4·85-s + 8·89-s − 12·95-s + 6·97-s + 2·101-s + 12·107-s − 4·113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.301·11-s − 0.485·17-s − 1.37·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s − 0.328·37-s + 0.312·41-s − 0.914·43-s − 0.875·47-s − 49-s − 0.274·53-s − 0.269·55-s + 1.02·61-s − 1.46·67-s + 0.712·71-s + 0.702·73-s − 1.35·79-s + 1.31·83-s − 0.433·85-s + 0.847·89-s − 1.23·95-s + 0.609·97-s + 0.199·101-s + 1.16·107-s − 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.345191708643617956158106330184, −7.64169774921111749547569285208, −6.54293323680220475986381969205, −6.17342328550991466807414813888, −5.28382917315601766898618434233, −4.46240446556071889689804551118, −3.53706427119729610002870049485, −2.33177257604628276146265779832, −1.74810519198558425601842381645, 0,
1.74810519198558425601842381645, 2.33177257604628276146265779832, 3.53706427119729610002870049485, 4.46240446556071889689804551118, 5.28382917315601766898618434233, 6.17342328550991466807414813888, 6.54293323680220475986381969205, 7.64169774921111749547569285208, 8.345191708643617956158106330184
