| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 4·7-s + 3·8-s + 9-s − 12-s + 4·14-s − 16-s − 2·17-s − 18-s − 19-s − 4·21-s + 8·23-s + 3·24-s + 27-s + 4·28-s + 2·29-s − 8·31-s − 5·32-s + 2·34-s − 36-s + 2·37-s + 38-s + 2·41-s + 4·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.872·21-s + 1.66·23-s + 0.612·24-s + 0.192·27-s + 0.755·28-s + 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s − 1/6·36-s + 0.328·37-s + 0.162·38-s + 0.312·41-s + 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.235692362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235692362\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.93052155006755, −12.71539765411960, −12.10140644678534, −11.22962928745314, −10.95142623392004, −10.44506730091531, −9.796938036024791, −9.667236740023965, −9.105223183450264, −8.819297888995773, −8.400909966401974, −7.749798087513000, −7.189501207604750, −6.930637276438148, −6.372757107680719, −5.735899040984754, −5.119102669637005, −4.611575814523644, −3.978104832563054, −3.507370382163851, −3.048408601269119, −2.395349047741549, −1.765825351911905, −0.9162619668593431, −0.4149877036933776,
0.4149877036933776, 0.9162619668593431, 1.765825351911905, 2.395349047741549, 3.048408601269119, 3.507370382163851, 3.978104832563054, 4.611575814523644, 5.119102669637005, 5.735899040984754, 6.372757107680719, 6.930637276438148, 7.189501207604750, 7.749798087513000, 8.400909966401974, 8.819297888995773, 9.105223183450264, 9.667236740023965, 9.796938036024791, 10.44506730091531, 10.95142623392004, 11.22962928745314, 12.10140644678534, 12.71539765411960, 12.93052155006755
