| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s − 2·9-s + 10-s − 12-s + 13-s − 14-s + 15-s + 16-s + 2·18-s + 4·19-s − 20-s − 21-s − 4·23-s + 24-s + 25-s − 26-s + 5·27-s + 28-s − 2·29-s − 30-s + 8·31-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 − 2/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.471·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.962·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−16.40969326971862, −15.80216342066804, −15.31269743960033, −14.76524140195754, −13.95189986438204, −13.69808142637341, −12.71220331856734, −12.08213590865895, −11.61593874098150, −11.34284606157409, −10.67491580415552, −10.02013730466639, −9.551580272496767, −8.691408236617334, −8.150220360817124, −7.901761292775841, −6.864091114107134, −6.527705842544021, −5.665659270293035, −5.180487889805464, −4.386553282545156, −3.463422825550876, −2.831930548748932, −1.826443521832250, −0.9282028279896069, 0,
0.9282028279896069, 1.826443521832250, 2.831930548748932, 3.463422825550876, 4.386553282545156, 5.180487889805464, 5.665659270293035, 6.527705842544021, 6.864091114107134, 7.901761292775841, 8.150220360817124, 8.691408236617334, 9.551580272496767, 10.02013730466639, 10.67491580415552, 11.34284606157409, 11.61593874098150, 12.08213590865895, 12.71220331856734, 13.69808142637341, 13.95189986438204, 14.76524140195754, 15.31269743960033, 15.80216342066804, 16.40969326971862
