| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 6·11-s − 12-s + 2·13-s + 16-s − 17-s − 18-s + 4·19-s − 6·22-s + 5·23-s + 24-s − 2·26-s − 27-s − 2·31-s − 32-s − 6·33-s + 34-s + 36-s − 3·37-s − 4·38-s − 2·39-s + 5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.27·22-s + 1.04·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.359·31-s − 0.176·32-s − 1.04·33-s + 0.171·34-s + 1/6·36-s − 0.493·37-s − 0.648·38-s − 0.320·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.274574394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274574394\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−9.109480806766652134159179797687, −8.236172648580912518007354741539, −7.23992587981760317588198616085, −6.68657084906175109973094228187, −6.02543455681993560579491524235, −5.07960655642635955288710480794, −4.03131199406379791673102858407, −3.18697731135031892531165139908, −1.69989521813268761889361746891, −0.884857132071887046606476855213,
0.884857132071887046606476855213, 1.69989521813268761889361746891, 3.18697731135031892531165139908, 4.03131199406379791673102858407, 5.07960655642635955288710480794, 6.02543455681993560579491524235, 6.68657084906175109973094228187, 7.23992587981760317588198616085, 8.236172648580912518007354741539, 9.109480806766652134159179797687
