| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s + 13-s + 16-s − 6·17-s + 18-s + 4·19-s + 24-s + 26-s + 27-s + 6·29-s − 8·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 4·38-s + 39-s − 6·41-s − 8·43-s + 12·47-s + 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + 0.160·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.85113804408597, −13.59601276515697, −13.29490791621026, −12.64693070166897, −12.13115761622722, −11.72101124463798, −11.14858842296965, −10.61915838368019, −10.22868241829747, −9.535662940943450, −8.998778338927748, −8.582009704479193, −8.081459701486160, −7.248136803965928, −7.082801437718019, −6.447816125736839, −5.856470036880914, −5.177814468528907, −4.802875444819918, −4.001741534204932, −3.721962687104667, −2.955409781057496, −2.463496735897515, −1.795985323413887, −1.119173932186612, 0,
1.119173932186612, 1.795985323413887, 2.463496735897515, 2.955409781057496, 3.721962687104667, 4.001741534204932, 4.802875444819918, 5.177814468528907, 5.856470036880914, 6.447816125736839, 7.082801437718019, 7.248136803965928, 8.081459701486160, 8.582009704479193, 8.998778338927748, 9.535662940943450, 10.22868241829747, 10.61915838368019, 11.14858842296965, 11.72101124463798, 12.13115761622722, 12.64693070166897, 13.29490791621026, 13.59601276515697, 13.85113804408597
