| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 14-s + 15-s + 16-s + 17-s + 18-s − 8·19-s + 20-s + 21-s − 2·22-s − 4·23-s + 24-s + 25-s + 27-s + 28-s + 2·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.538852159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.538852159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 673 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.66985227985397, −12.24868866362964, −11.84982683998046, −11.05724301050924, −10.69067903234914, −10.54145036822078, −9.880024905273414, −9.309937085128107, −8.971220181899444, −8.317922562323608, −7.937712762538393, −7.572960750031394, −7.002330259831439, −6.335246602410793, −6.024477208078953, −5.635686314020283, −4.808265043862576, −4.539011420719420, −4.105068795255826, −3.430028119515920, −2.864772633155512, −2.396057388180233, −1.875145314164775, −1.443235991577817, −0.4273386074491847,
0.4273386074491847, 1.443235991577817, 1.875145314164775, 2.396057388180233, 2.864772633155512, 3.430028119515920, 4.105068795255826, 4.539011420719420, 4.808265043862576, 5.635686314020283, 6.024477208078953, 6.335246602410793, 7.002330259831439, 7.572960750031394, 7.937712762538393, 8.317922562323608, 8.971220181899444, 9.309937085128107, 9.880024905273414, 10.54145036822078, 10.69067903234914, 11.05724301050924, 11.84982683998046, 12.24868866362964, 12.66985227985397
