| L(s) = 1 | − 3-s − 5-s − 3·7-s − 2·9-s + 2·11-s + 13-s + 15-s − 3·17-s + 3·21-s − 4·23-s − 4·25-s + 5·27-s + 2·29-s + 4·31-s − 2·33-s + 3·35-s + 5·37-s − 39-s + 12·41-s + 7·43-s + 2·45-s + 9·47-s + 2·49-s + 3·51-s + 4·53-s − 2·55-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.654·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.507·35-s + 0.821·37-s − 0.160·39-s + 1.87·41-s + 1.06·43-s + 0.298·45-s + 1.31·47-s + 2/7·49-s + 0.420·51-s + 0.549·53-s − 0.269·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.79857909382524, −12.42963121324381, −11.91497256580094, −11.59237884733221, −11.20933831422595, −10.59925258518140, −10.32987563050355, −9.579947154356144, −9.301156449646962, −8.886464038252333, −8.165159108018741, −7.970649807991953, −7.057676552824404, −6.933669745197003, −6.131229674743856, −5.891612951265093, −5.710611950927409, −4.659901240474571, −4.231135974599457, −3.932031688754640, −3.204052999940965, −2.648470542722564, −2.252938239918532, −1.201420838604381, −0.6224847692165643, 0,
0.6224847692165643, 1.201420838604381, 2.252938239918532, 2.648470542722564, 3.204052999940965, 3.932031688754640, 4.231135974599457, 4.659901240474571, 5.710611950927409, 5.891612951265093, 6.131229674743856, 6.933669745197003, 7.057676552824404, 7.970649807991953, 8.165159108018741, 8.886464038252333, 9.301156449646962, 9.579947154356144, 10.32987563050355, 10.59925258518140, 11.20933831422595, 11.59237884733221, 11.91497256580094, 12.42963121324381, 12.79857909382524
