| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 2·11-s + 12-s − 13-s + 16-s + 17-s − 18-s − 4·19-s + 2·22-s − 24-s + 26-s + 27-s − 8·29-s + 8·31-s − 32-s − 2·33-s − 34-s + 36-s + 8·37-s + 4·38-s − 39-s − 6·41-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 − 0.603·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.426·22-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s + 1/6·36-s + 1.31·37-s + 0.648·38-s − 0.160·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−13.58032712382297, −13.40969706318217, −12.80705004478857, −12.28899737837294, −11.84292740686432, −11.26031311376793, −10.66619037681307, −10.40477849089746, −9.736019852346449, −9.469347543084049, −8.791712416458760, −8.446633211250459, −7.841648712195058, −7.549556615778007, −7.005725754198461, −6.290465064860946, −5.964756226812439, −5.164306844352951, −4.643052297514633, −3.964366534177557, −3.402449084108405, −2.634783737582203, −2.324183066245309, −1.612570007872358, −0.8204313166003201, 0,
0.8204313166003201, 1.612570007872358, 2.324183066245309, 2.634783737582203, 3.402449084108405, 3.964366534177557, 4.643052297514633, 5.164306844352951, 5.964756226812439, 6.290465064860946, 7.005725754198461, 7.549556615778007, 7.841648712195058, 8.446633211250459, 8.791712416458760, 9.469347543084049, 9.736019852346449, 10.40477849089746, 10.66619037681307, 11.26031311376793, 11.84292740686432, 12.28899737837294, 12.80705004478857, 13.40969706318217, 13.58032712382297
