| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 11-s − 12-s − 13-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s + 22-s − 4·23-s − 24-s + 25-s − 26-s − 27-s − 4·29-s − 30-s − 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.213·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.742·29-s − 0.182·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.25226347223480, −12.75497434965405, −12.20413756698703, −11.89101414784218, −11.68467592633531, −10.88158829228970, −10.55243631217614, −9.926426823917093, −9.756238557083473, −9.100287219390586, −8.515887982278393, −7.732608133999368, −7.527098206449147, −6.964786810013763, −6.329142798700608, −5.963080963025938, −5.462533530663844, −5.047957972613366, −4.636621296576931, −3.782508207463108, −3.411691288053931, −2.960861684644654, −1.938984524383624, −1.675360362088727, −0.9233973144518872, 0,
0.9233973144518872, 1.675360362088727, 1.938984524383624, 2.960861684644654, 3.411691288053931, 3.782508207463108, 4.636621296576931, 5.047957972613366, 5.462533530663844, 5.963080963025938, 6.329142798700608, 6.964786810013763, 7.527098206449147, 7.732608133999368, 8.515887982278393, 9.100287219390586, 9.756238557083473, 9.926426823917093, 10.55243631217614, 10.88158829228970, 11.68467592633531, 11.89101414784218, 12.20413756698703, 12.75497434965405, 13.25226347223480
