| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 4·11-s + 13-s + 16-s + 2·17-s − 4·19-s − 20-s − 4·22-s − 4·23-s + 25-s + 26-s − 2·29-s + 8·31-s + 32-s + 2·34-s − 6·37-s − 4·38-s − 40-s + 10·41-s − 4·43-s − 4·44-s − 4·46-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.986·37-s − 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.589·46-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.53163852378278, −14.18706059097619, −13.37355181107495, −13.27951032849569, −12.62274472206506, −12.02065390872981, −11.83343200483024, −11.01545288542987, −10.57798266638228, −10.23836764719469, −9.594356270811414, −8.754039741811760, −8.296756739705971, −7.786008653823660, −7.336115073663466, −6.654686450607162, −5.991080479697107, −5.660254845529507, −4.836329743886886, −4.482613703848471, −3.767072711411085, −3.217610745697672, −2.505490963485529, −1.981299038353714, −0.9442174890710293, 0,
0.9442174890710293, 1.981299038353714, 2.505490963485529, 3.217610745697672, 3.767072711411085, 4.482613703848471, 4.836329743886886, 5.660254845529507, 5.991080479697107, 6.654686450607162, 7.336115073663466, 7.786008653823660, 8.296756739705971, 8.754039741811760, 9.594356270811414, 10.23836764719469, 10.57798266638228, 11.01545288542987, 11.83343200483024, 12.02065390872981, 12.62274472206506, 13.27951032849569, 13.37355181107495, 14.18706059097619, 14.53163852378278
