| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 12-s + 15-s + 16-s − 3·17-s + 18-s + 19-s + 20-s + 24-s + 25-s + 27-s + 6·29-s + 30-s − 2·31-s + 32-s − 3·34-s + 36-s + 10·37-s + 38-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 1/6·36-s + 1.64·37-s + 0.162·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.677597991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.677597991\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.99997456711529, −12.55289278610959, −12.00171021622397, −11.52181170871711, −11.09911680815831, −10.48301292831623, −10.18318602179962, −9.551463365734909, −9.208964983168755, −8.574762399426557, −8.230521471174606, −7.615761527708651, −7.102067094687956, −6.664523305377065, −6.199261147297358, −5.626674074485523, −5.148351082057565, −4.564346219085860, −4.108822693955475, −3.608638703333051, −2.941163635296493, −2.388322354787284, −2.134971125868042, −1.222947788146643, −0.6628911809894992,
0.6628911809894992, 1.222947788146643, 2.134971125868042, 2.388322354787284, 2.941163635296493, 3.608638703333051, 4.108822693955475, 4.564346219085860, 5.148351082057565, 5.626674074485523, 6.199261147297358, 6.664523305377065, 7.102067094687956, 7.615761527708651, 8.230521471174606, 8.574762399426557, 9.208964983168755, 9.551463365734909, 10.18318602179962, 10.48301292831623, 11.09911680815831, 11.52181170871711, 12.00171021622397, 12.55289278610959, 12.99997456711529
