| L(s) = 1 | − 4·7-s + 11-s + 4·13-s + 6·17-s − 2·19-s + 6·23-s − 5·25-s − 6·29-s − 4·31-s − 37-s − 6·41-s − 2·43-s + 9·49-s + 6·53-s + 6·59-s − 8·61-s + 4·67-s + 12·71-s + 2·73-s − 4·77-s + 2·79-s − 12·83-s − 16·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.301·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s − 25-s − 1.11·29-s − 0.718·31-s − 0.164·37-s − 0.937·41-s − 0.304·43-s + 9/7·49-s + 0.824·53-s + 0.781·59-s − 1.02·61-s + 0.488·67-s + 1.42·71-s + 0.234·73-s − 0.455·77-s + 0.225·79-s − 1.31·83-s − 1.67·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.13481545924209, −12.73363007590393, −12.29412482404783, −11.82358273864105, −11.22714222451079, −10.85870752270925, −10.29433488087845, −9.814665393730587, −9.454617437818906, −9.061234480611235, −8.440295028505688, −8.072849383713186, −7.246372546976085, −7.054827256846256, −6.427837359232978, −5.983587614366806, −5.542298998276604, −5.117574386872385, −4.149899669912966, −3.689460705625222, −3.411529815689846, −2.908283018825535, −2.080042554545946, −1.431806381870125, −0.7579417554609899, 0,
0.7579417554609899, 1.431806381870125, 2.080042554545946, 2.908283018825535, 3.411529815689846, 3.689460705625222, 4.149899669912966, 5.117574386872385, 5.542298998276604, 5.983587614366806, 6.427837359232978, 7.054827256846256, 7.246372546976085, 8.072849383713186, 8.440295028505688, 9.061234480611235, 9.454617437818906, 9.814665393730587, 10.29433488087845, 10.85870752270925, 11.22714222451079, 11.82358273864105, 12.29412482404783, 12.73363007590393, 13.13481545924209
