| L(s) = 1 | + 2·5-s + 4·7-s − 4·11-s + 2·17-s − 23-s − 25-s + 2·29-s + 8·35-s + 10·37-s − 6·41-s + 8·43-s − 8·47-s + 9·49-s + 6·53-s − 8·55-s − 4·59-s + 14·61-s − 8·67-s − 8·71-s + 6·73-s − 16·77-s + 12·79-s − 12·83-s + 4·85-s − 2·89-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.485·17-s − 0.208·23-s − 1/5·25-s + 0.371·29-s + 1.35·35-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 1.07·55-s − 0.520·59-s + 1.79·61-s − 0.977·67-s − 0.949·71-s + 0.702·73-s − 1.82·77-s + 1.35·79-s − 1.31·83-s + 0.433·85-s − 0.211·89-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 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
−13.10661343692018, −12.52626143818114, −12.02244435387772, −11.56782731455667, −11.09730804614254, −10.68849865905318, −10.25750256330460, −9.767088519704401, −9.431893938121395, −8.682459851055352, −8.295658689079600, −7.792473318676646, −7.647698662462391, −6.855785061438338, −6.334030371977706, −5.651861168410147, −5.412284380309035, −5.016333371520833, −4.360576989984413, −3.978955655836383, −3.031297886941099, −2.544628326542478, −2.111020817706697, −1.470701535588566, −0.9830052316594001, 0,
0.9830052316594001, 1.470701535588566, 2.111020817706697, 2.544628326542478, 3.031297886941099, 3.978955655836383, 4.360576989984413, 5.016333371520833, 5.412284380309035, 5.651861168410147, 6.334030371977706, 6.855785061438338, 7.647698662462391, 7.792473318676646, 8.295658689079600, 8.682459851055352, 9.431893938121395, 9.767088519704401, 10.25750256330460, 10.68849865905318, 11.09730804614254, 11.56782731455667, 12.02244435387772, 12.52626143818114, 13.10661343692018
