| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s + 4·11-s − 12-s − 4·13-s + 16-s + 2·17-s − 2·18-s + 6·19-s + 4·22-s + 23-s − 24-s − 4·26-s + 5·27-s + 3·29-s − 4·31-s + 32-s − 4·33-s + 2·34-s − 2·36-s − 6·37-s + 6·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.37·19-s + 0.852·22-s + 0.208·23-s − 0.204·24-s − 0.784·26-s + 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.176·32-s − 0.696·33-s + 0.342·34-s − 1/3·36-s − 0.986·37-s + 0.973·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.613968262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.613968262\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−13.01411204910567, −12.42924820848376, −12.13381900149517, −11.78835732840900, −11.39082626591793, −10.97630415387174, −10.23802008052655, −9.947126317292862, −9.331453649396228, −8.887269900418303, −8.314643183426327, −7.622218349093591, −7.230017836793160, −6.703199739049858, −6.319488548102611, −5.634367651257363, −5.249392479535112, −4.938032212995759, −4.264837140772695, −3.489974261730375, −3.314186321004526, −2.518442248058041, −1.926463223172612, −1.113440228986848, −0.5431755283029146,
0.5431755283029146, 1.113440228986848, 1.926463223172612, 2.518442248058041, 3.314186321004526, 3.489974261730375, 4.264837140772695, 4.938032212995759, 5.249392479535112, 5.634367651257363, 6.319488548102611, 6.703199739049858, 7.230017836793160, 7.622218349093591, 8.314643183426327, 8.887269900418303, 9.331453649396228, 9.947126317292862, 10.23802008052655, 10.97630415387174, 11.39082626591793, 11.78835732840900, 12.13381900149517, 12.42924820848376, 13.01411204910567
