| L(s) = 1 | − 2·3-s + 5-s − 3·7-s + 9-s + 5·11-s − 2·15-s − 3·17-s − 19-s + 6·21-s − 8·23-s − 4·25-s + 4·27-s − 2·29-s + 4·31-s − 10·33-s − 3·35-s − 10·37-s − 10·41-s − 43-s + 45-s − 47-s + 2·49-s + 6·51-s − 4·53-s + 5·55-s + 2·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 0.516·15-s − 0.727·17-s − 0.229·19-s + 1.30·21-s − 1.66·23-s − 4/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 1.74·33-s − 0.507·35-s − 1.64·37-s − 1.56·41-s − 0.152·43-s + 0.149·45-s − 0.145·47-s + 2/7·49-s + 0.840·51-s − 0.549·53-s + 0.674·55-s + 0.264·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.09993971138493, −14.31582584788097, −13.97317968399640, −13.42688893531958, −12.95935500578103, −12.14574700413929, −11.97391935080462, −11.63493975938571, −10.86488407266180, −10.35039542515102, −9.845886664502317, −9.514734363101855, −8.724666747640242, −8.431397854512566, −7.397527345356928, −6.693794156718488, −6.542714650909938, −5.907357916305928, −5.677845731208944, −4.720861535673048, −4.195173331231536, −3.570704586465669, −2.902552989166456, −1.869571408199462, −1.406128016671732, 0, 0,
1.406128016671732, 1.869571408199462, 2.902552989166456, 3.570704586465669, 4.195173331231536, 4.720861535673048, 5.677845731208944, 5.907357916305928, 6.542714650909938, 6.693794156718488, 7.397527345356928, 8.431397854512566, 8.724666747640242, 9.514734363101855, 9.845886664502317, 10.35039542515102, 10.86488407266180, 11.63493975938571, 11.97391935080462, 12.14574700413929, 12.95935500578103, 13.42688893531958, 13.97317968399640, 14.31582584788097, 15.09993971138493
