| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s + 4·19-s + 25-s − 27-s + 2·29-s + 4·33-s + 10·37-s − 2·39-s + 10·41-s + 4·43-s − 45-s − 8·47-s − 7·49-s − 2·51-s + 10·53-s + 4·55-s − 4·57-s − 4·59-s + 2·61-s − 2·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.64·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s + 1.37·53-s + 0.539·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.248·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.128713699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128713699\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−10.09004766089519189801221596346, −9.339349995423015885465559763611, −8.074746559817531846840486780032, −7.67749235610857150327119620769, −6.56365355473852823699216167747, −5.63406482120703248001185629214, −4.87700802399738744801917392626, −3.76558570458913747879946495025, −2.61544872003966305633910980827, −0.870154925974925884967934945032,
0.870154925974925884967934945032, 2.61544872003966305633910980827, 3.76558570458913747879946495025, 4.87700802399738744801917392626, 5.63406482120703248001185629214, 6.56365355473852823699216167747, 7.67749235610857150327119620769, 8.074746559817531846840486780032, 9.339349995423015885465559763611, 10.09004766089519189801221596346
