| L(s) = 1 | + 2·5-s + 4·11-s − 13-s + 2·17-s − 25-s + 10·29-s + 4·31-s − 2·37-s + 6·41-s + 12·43-s − 6·53-s + 8·55-s − 12·59-s + 2·61-s − 2·65-s + 8·67-s − 2·73-s − 8·79-s − 4·83-s + 4·85-s − 2·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s − 0.277·13-s + 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.82·43-s − 0.824·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s − 0.234·73-s − 0.900·79-s − 0.439·83-s + 0.433·85-s − 0.211·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.991796379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.991796379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 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
−13.88310958211594, −13.58347660970836, −12.69474892774289, −12.43076509511829, −11.97965756824696, −11.40287719167841, −10.84313378485852, −10.32635825278756, −9.741739138851306, −9.499571013753101, −8.943934408737983, −8.379120947390383, −7.825427192706436, −7.199451374475913, −6.595976193093704, −6.201741202556586, −5.738355394531530, −5.119679693609449, −4.343198842924883, −4.117108076146181, −3.091164119393045, −2.726538970046844, −1.912479404400066, −1.295949686654049, −0.6791748441924608,
0.6791748441924608, 1.295949686654049, 1.912479404400066, 2.726538970046844, 3.091164119393045, 4.117108076146181, 4.343198842924883, 5.119679693609449, 5.738355394531530, 6.201741202556586, 6.595976193093704, 7.199451374475913, 7.825427192706436, 8.379120947390383, 8.943934408737983, 9.499571013753101, 9.741739138851306, 10.32635825278756, 10.84313378485852, 11.40287719167841, 11.97965756824696, 12.43076509511829, 12.69474892774289, 13.58347660970836, 13.88310958211594
