| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 9-s − 6·11-s + 4·15-s + 17-s − 4·21-s − 6·23-s − 25-s − 4·27-s − 10·29-s + 2·31-s − 12·33-s − 4·35-s − 6·37-s + 6·41-s + 8·43-s + 2·45-s − 3·49-s + 2·51-s − 10·53-s − 12·55-s − 8·59-s + 14·61-s − 2·63-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 1.03·15-s + 0.242·17-s − 0.872·21-s − 1.25·23-s − 1/5·25-s − 0.769·27-s − 1.85·29-s + 0.359·31-s − 2.08·33-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.298·45-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 1.61·55-s − 1.04·59-s + 1.79·61-s − 0.251·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926508984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926508984\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.47932488646357, −13.97133892947660, −13.71316919572247, −13.13898499984855, −12.75312678682118, −12.37739894575498, −11.38699399480799, −10.88244386631130, −10.25516431878633, −9.786510456550806, −9.457907971116700, −8.946004382258075, −8.201866134312671, −7.694969397352412, −7.527941767189343, −6.448947918126679, −5.999611282963110, −5.434861340638060, −4.932662203523629, −3.814840632648145, −3.517226474434911, −2.689239129166747, −2.258780874957687, −1.813061107153724, −0.4157484148391295,
0.4157484148391295, 1.813061107153724, 2.258780874957687, 2.689239129166747, 3.517226474434911, 3.814840632648145, 4.932662203523629, 5.434861340638060, 5.999611282963110, 6.448947918126679, 7.527941767189343, 7.694969397352412, 8.201866134312671, 8.946004382258075, 9.457907971116700, 9.786510456550806, 10.25516431878633, 10.88244386631130, 11.38699399480799, 12.37739894575498, 12.75312678682118, 13.13898499984855, 13.71316919572247, 13.97133892947660, 14.47932488646357
