| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 2·13-s + 15-s − 6·17-s − 4·19-s − 21-s − 8·23-s + 25-s − 27-s − 6·29-s + 4·31-s + 33-s − 35-s + 2·37-s − 2·39-s − 6·41-s + 12·43-s − 45-s + 49-s + 6·51-s + 2·53-s + 55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s + 0.274·53-s + 0.134·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9091010731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9091010731\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.70100095091290825022096262556, −7.03471564348879185663033571990, −6.19845162855533512646462735970, −5.86260375849636690024930073817, −4.81484967856810204655145614778, −4.27784002976064669754156595151, −3.70448934051116977633409995430, −2.44920194318402002902520351720, −1.76954475875817266771400278922, −0.46033672039814631545460229320,
0.46033672039814631545460229320, 1.76954475875817266771400278922, 2.44920194318402002902520351720, 3.70448934051116977633409995430, 4.27784002976064669754156595151, 4.81484967856810204655145614778, 5.86260375849636690024930073817, 6.19845162855533512646462735970, 7.03471564348879185663033571990, 7.70100095091290825022096262556
