| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 3·7-s − 8-s − 2·9-s + 12-s − 3·14-s + 16-s − 8·17-s + 2·18-s + 8·19-s + 3·21-s − 24-s − 5·27-s + 3·28-s + 2·29-s + 6·31-s − 32-s + 8·34-s − 2·36-s − 8·38-s + 5·41-s − 3·42-s − 43-s + 5·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.801·14-s + 1/4·16-s − 1.94·17-s + 0.471·18-s + 1.83·19-s + 0.654·21-s − 0.204·24-s − 0.962·27-s + 0.566·28-s + 0.371·29-s + 1.07·31-s − 0.176·32-s + 1.37·34-s − 1/3·36-s − 1.29·38-s + 0.780·41-s − 0.462·42-s − 0.152·43-s + 0.729·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.836799982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836799982\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.975397461043819242049289575936, −7.76845975296967096403534094138, −6.81293444839265906996753729263, −6.08155943292101122318971721354, −5.14045471063472899790242596654, −4.52144076268887445366930213235, −3.41606230007644160685535869829, −2.57669650865983731851662402778, −1.89518494278143891156708444310, −0.77044182772292435940890208695,
0.77044182772292435940890208695, 1.89518494278143891156708444310, 2.57669650865983731851662402778, 3.41606230007644160685535869829, 4.52144076268887445366930213235, 5.14045471063472899790242596654, 6.08155943292101122318971721354, 6.81293444839265906996753729263, 7.76845975296967096403534094138, 7.975397461043819242049289575936
