| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 14-s − 15-s + 16-s − 6·17-s + 18-s + 8·19-s + 20-s − 21-s − 4·22-s + 8·23-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 − 1.20·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s + 1.66·23-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.279710665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.279710665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 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.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.17958212448441, −14.28972656005383, −13.71564626239480, −13.41587097701237, −12.85890420078930, −12.47152972977303, −11.68551423980736, −11.26903476128730, −10.83614464804452, −10.36348497905595, −9.708335103575741, −9.030261197657245, −8.535063985395077, −7.632069270718977, −7.121704824214699, −6.853729417700670, −5.864359925832705, −5.418795006525001, −5.068625197876565, −4.510631568356031, −3.658964778773060, −2.898539634713534, −2.342563699402519, −1.509118656150954, −0.6261281006863611,
0.6261281006863611, 1.509118656150954, 2.342563699402519, 2.898539634713534, 3.658964778773060, 4.510631568356031, 5.068625197876565, 5.418795006525001, 5.864359925832705, 6.853729417700670, 7.121704824214699, 7.632069270718977, 8.535063985395077, 9.030261197657245, 9.708335103575741, 10.36348497905595, 10.83614464804452, 11.26903476128730, 11.68551423980736, 12.47152972977303, 12.85890420078930, 13.41587097701237, 13.71564626239480, 14.28972656005383, 15.17958212448441
