| L(s) = 1 | + 3·5-s − 3·7-s − 11-s + 2·13-s − 3·17-s − 19-s + 4·25-s + 4·29-s − 9·35-s + 37-s + 6·41-s + 3·43-s − 11·47-s + 2·49-s + 2·53-s − 3·55-s − 6·59-s + 5·61-s + 6·65-s − 6·67-s − 10·71-s − 11·73-s + 3·77-s + 14·79-s − 12·83-s − 9·85-s − 4·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 0.301·11-s + 0.554·13-s − 0.727·17-s − 0.229·19-s + 4/5·25-s + 0.742·29-s − 1.52·35-s + 0.164·37-s + 0.937·41-s + 0.457·43-s − 1.60·47-s + 2/7·49-s + 0.274·53-s − 0.404·55-s − 0.781·59-s + 0.640·61-s + 0.744·65-s − 0.733·67-s − 1.18·71-s − 1.28·73-s + 0.341·77-s + 1.57·79-s − 1.31·83-s − 0.976·85-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045288741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045288741\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.48500723383487, −13.40699552924740, −12.82834292763739, −12.54809175632916, −11.79961957132505, −11.19713118872057, −10.70112714015471, −10.19012909463556, −9.823004055783876, −9.364065802663341, −8.882472738801338, −8.429918956144347, −7.710575264069102, −7.020696724778613, −6.546162966561802, −6.065968503853882, −5.844023349670096, −5.075885039164947, −4.476980074893985, −3.856264831242554, −2.995891445104802, −2.748732962301131, −1.962822866753258, −1.375613291833492, −0.4369822049892321,
0.4369822049892321, 1.375613291833492, 1.962822866753258, 2.748732962301131, 2.995891445104802, 3.856264831242554, 4.476980074893985, 5.075885039164947, 5.844023349670096, 6.065968503853882, 6.546162966561802, 7.020696724778613, 7.710575264069102, 8.429918956144347, 8.882472738801338, 9.364065802663341, 9.823004055783876, 10.19012909463556, 10.70112714015471, 11.19713118872057, 11.79961957132505, 12.54809175632916, 12.82834292763739, 13.40699552924740, 13.48500723383487
