| L(s) = 1 | − 4·7-s − 11-s + 4·13-s − 2·17-s + 2·19-s − 2·23-s − 5·25-s − 10·29-s + 8·31-s + 37-s + 6·41-s − 6·43-s − 4·47-s + 9·49-s + 6·53-s − 10·59-s − 12·67-s − 8·71-s + 10·73-s + 4·77-s − 2·79-s − 12·83-s − 12·89-s − 16·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s − 0.417·23-s − 25-s − 1.85·29-s + 1.43·31-s + 0.164·37-s + 0.937·41-s − 0.914·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 1.30·59-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 0.225·79-s − 1.31·83-s − 1.27·89-s − 1.67·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8374439457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8374439457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−14.12986203291451, −13.74687912223144, −13.21461574919437, −13.03137476102822, −12.40657942088953, −11.72532080528874, −11.37432941308244, −10.73913180761161, −10.12485103744218, −9.759488042088487, −9.246603644105032, −8.740846410627421, −8.104890830776871, −7.512851131757117, −6.987580557812617, −6.245478322712705, −6.016315895910012, −5.498937939830032, −4.533570313070153, −3.990855443736130, −3.389466724512781, −2.935643248781384, −2.125329518959127, −1.331759506767251, −0.3162845598533506,
0.3162845598533506, 1.331759506767251, 2.125329518959127, 2.935643248781384, 3.389466724512781, 3.990855443736130, 4.533570313070153, 5.498937939830032, 6.016315895910012, 6.245478322712705, 6.987580557812617, 7.512851131757117, 8.104890830776871, 8.740846410627421, 9.246603644105032, 9.759488042088487, 10.12485103744218, 10.73913180761161, 11.37432941308244, 11.72532080528874, 12.40657942088953, 13.03137476102822, 13.21461574919437, 13.74687912223144, 14.12986203291451
