| L(s) = 1 | + 3·3-s − 5-s − 7-s + 6·9-s − 2·11-s − 3·15-s − 3·17-s + 6·19-s − 3·21-s − 4·23-s − 4·25-s + 9·27-s − 2·29-s − 4·31-s − 6·33-s + 35-s + 3·37-s + 5·43-s − 6·45-s − 13·47-s − 6·49-s − 9·51-s − 12·53-s + 2·55-s + 18·57-s − 10·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 0.603·11-s − 0.774·15-s − 0.727·17-s + 1.37·19-s − 0.654·21-s − 0.834·23-s − 4/5·25-s + 1.73·27-s − 0.371·29-s − 0.718·31-s − 1.04·33-s + 0.169·35-s + 0.493·37-s + 0.762·43-s − 0.894·45-s − 1.89·47-s − 6/7·49-s − 1.26·51-s − 1.64·53-s + 0.269·55-s + 2.38·57-s − 1.30·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−16.46136795649626, −16.01375199288389, −15.63091047140781, −15.12016126894525, −14.49200429748958, −13.89009095450240, −13.61018544864798, −12.78664723720726, −12.59075349787387, −11.51561958845496, −11.08701501109741, −10.09885137884058, −9.545522687466834, −9.337673611302506, −8.372504302205781, −7.952773675345803, −7.549275187558091, −6.837934892432402, −5.992691282957394, −5.032529861838328, −4.257701026641828, −3.540702448211370, −3.090632834420291, −2.278417408877418, −1.529417331846637, 0,
1.529417331846637, 2.278417408877418, 3.090632834420291, 3.540702448211370, 4.257701026641828, 5.032529861838328, 5.992691282957394, 6.837934892432402, 7.549275187558091, 7.952773675345803, 8.372504302205781, 9.337673611302506, 9.545522687466834, 10.09885137884058, 11.08701501109741, 11.51561958845496, 12.59075349787387, 12.78664723720726, 13.61018544864798, 13.89009095450240, 14.49200429748958, 15.12016126894525, 15.63091047140781, 16.01375199288389, 16.46136795649626
