| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 2·11-s − 2·13-s − 15-s − 2·17-s − 2·19-s − 2·21-s − 2·23-s + 25-s + 27-s − 6·29-s − 4·31-s + 2·33-s + 2·35-s + 2·37-s − 2·39-s − 10·41-s + 8·43-s − 45-s − 2·47-s − 3·49-s − 2·51-s + 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.436·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.348·33-s + 0.338·35-s + 0.328·37-s − 0.320·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.935666680146898108947197436754, −8.037322825083629513883200764970, −7.24670428133688603846896880841, −6.60668142823229757211620489443, −5.66052745164546366174049172250, −4.46902289837185323150742647792, −3.77169781021970174248128836681, −2.88684999428318360090105085232, −1.76828397665908917268536961976, 0,
1.76828397665908917268536961976, 2.88684999428318360090105085232, 3.77169781021970174248128836681, 4.46902289837185323150742647792, 5.66052745164546366174049172250, 6.60668142823229757211620489443, 7.24670428133688603846896880841, 8.037322825083629513883200764970, 8.935666680146898108947197436754
