| L(s) = 1 | + 3-s − 3·5-s − 7-s − 2·9-s − 5·13-s − 3·15-s − 3·17-s − 21-s + 3·23-s + 4·25-s − 5·27-s − 3·29-s + 8·31-s + 3·35-s − 2·37-s − 5·39-s + 3·41-s + 43-s + 6·45-s + 9·47-s + 49-s − 3·51-s − 3·53-s + 9·59-s − 7·61-s + 2·63-s + 15·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 1.38·13-s − 0.774·15-s − 0.727·17-s − 0.218·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.557·29-s + 1.43·31-s + 0.507·35-s − 0.328·37-s − 0.800·39-s + 0.468·41-s + 0.152·43-s + 0.894·45-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 0.412·53-s + 1.17·59-s − 0.896·61-s + 0.251·63-s + 1.86·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−14.92623037834850, −14.78170481243717, −13.92158180564169, −13.66234489047458, −12.83450752682560, −12.42720186640539, −11.88305024724723, −11.47423276725354, −10.94749641018408, −10.31388813213465, −9.648152153962496, −9.087504908358294, −8.698507062296115, −7.896492442474477, −7.776491286754906, −6.993705773980753, −6.616742908964949, −5.672537166407992, −5.102305495595044, −4.305607331669766, −3.998208273890279, −3.072583824278691, −2.761535879891882, −2.036296300523445, −0.7211986534966410, 0,
0.7211986534966410, 2.036296300523445, 2.761535879891882, 3.072583824278691, 3.998208273890279, 4.305607331669766, 5.102305495595044, 5.672537166407992, 6.616742908964949, 6.993705773980753, 7.776491286754906, 7.896492442474477, 8.698507062296115, 9.087504908358294, 9.648152153962496, 10.31388813213465, 10.94749641018408, 11.47423276725354, 11.88305024724723, 12.42720186640539, 12.83450752682560, 13.66234489047458, 13.92158180564169, 14.78170481243717, 14.92623037834850
