| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 4·11-s + 2·14-s + 16-s + 4·19-s − 4·22-s − 4·23-s − 2·28-s + 6·29-s + 8·31-s − 32-s − 6·37-s − 4·38-s + 8·41-s − 2·43-s + 4·44-s + 4·46-s − 8·47-s − 3·49-s + 14·53-s + 2·56-s − 6·58-s − 6·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.20·11-s + 0.534·14-s + 1/4·16-s + 0.917·19-s − 0.852·22-s − 0.834·23-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.986·37-s − 0.648·38-s + 1.24·41-s − 0.304·43-s + 0.603·44-s + 0.589·46-s − 1.16·47-s − 3/7·49-s + 1.92·53-s + 0.267·56-s − 0.787·58-s − 0.781·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628102001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628102001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.62801632650321, −12.84022357760877, −12.47397480725204, −11.79524229268824, −11.72067951932245, −11.14930264331839, −10.24674548141654, −10.09512586151089, −9.711318723500237, −9.055982801551007, −8.683480709678984, −8.233799621863323, −7.503823194983494, −7.133693752778951, −6.490331207585655, −6.208199447074663, −5.689555948485121, −4.847267325922907, −4.305268338948574, −3.652041366863532, −3.091282681776535, −2.604256550199439, −1.720313577749624, −1.156281660048516, −0.4761298457945450,
0.4761298457945450, 1.156281660048516, 1.720313577749624, 2.604256550199439, 3.091282681776535, 3.652041366863532, 4.305268338948574, 4.847267325922907, 5.689555948485121, 6.208199447074663, 6.490331207585655, 7.133693752778951, 7.503823194983494, 8.233799621863323, 8.683480709678984, 9.055982801551007, 9.711318723500237, 10.09512586151089, 10.24674548141654, 11.14930264331839, 11.72067951932245, 11.79524229268824, 12.47397480725204, 12.84022357760877, 13.62801632650321
