| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s + 6·13-s + 2·17-s − 2·21-s + 4·23-s + 27-s + 8·31-s − 2·33-s + 2·37-s + 6·39-s + 2·41-s + 4·43-s + 8·47-s − 3·49-s + 2·51-s + 6·53-s − 10·59-s + 2·61-s − 2·63-s + 8·67-s + 4·69-s − 12·71-s − 4·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.485·17-s − 0.436·21-s + 0.834·23-s + 0.192·27-s + 1.43·31-s − 0.348·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 1.30·59-s + 0.256·61-s − 0.251·63-s + 0.977·67-s + 0.481·69-s − 1.42·71-s − 0.468·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990547316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990547316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−9.692635690112158052669440441100, −8.876429119746077603127030187363, −8.236259450125909594320955817325, −7.34830902118093822137916374938, −6.39955394184406631525545593399, −5.66765515312328094809085573246, −4.40421151945420287303938949967, −3.43332661175523197849238647019, −2.66426897593599555100997020524, −1.09380620717066232730427473522,
1.09380620717066232730427473522, 2.66426897593599555100997020524, 3.43332661175523197849238647019, 4.40421151945420287303938949967, 5.66765515312328094809085573246, 6.39955394184406631525545593399, 7.34830902118093822137916374938, 8.236259450125909594320955817325, 8.876429119746077603127030187363, 9.692635690112158052669440441100
