| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s − 11-s + 12-s − 2·13-s + 2·14-s + 16-s + 6·17-s + 18-s + 2·21-s − 22-s − 4·23-s + 24-s − 5·25-s − 2·26-s + 27-s + 2·28-s − 2·29-s − 8·31-s + 32-s − 33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.436·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.00241340406600, −13.50325074032404, −12.87981581177202, −12.53753519468169, −12.02057930648537, −11.55892668881890, −11.09759910514984, −10.40988607540399, −10.07983176262021, −9.483764034134929, −9.023563961177192, −8.208082069354267, −7.859573683782530, −7.501626608395888, −7.047242738454330, −6.181868987484996, −5.631357492401854, −5.339691370542377, −4.602505146474994, −4.141216505211474, −3.478339346438037, −3.098558673503377, −2.176627741991116, −1.897150844281683, −1.128040188796757, 0,
1.128040188796757, 1.897150844281683, 2.176627741991116, 3.098558673503377, 3.478339346438037, 4.141216505211474, 4.602505146474994, 5.339691370542377, 5.631357492401854, 6.181868987484996, 7.047242738454330, 7.501626608395888, 7.859573683782530, 8.208082069354267, 9.023563961177192, 9.483764034134929, 10.07983176262021, 10.40988607540399, 11.09759910514984, 11.55892668881890, 12.02057930648537, 12.53753519468169, 12.87981581177202, 13.50325074032404, 14.00241340406600
