| L(s) = 1 | + 2-s + 4-s − 2·5-s − 4·7-s + 8-s − 2·10-s + 11-s + 6·13-s − 4·14-s + 16-s − 2·17-s − 2·20-s + 22-s − 4·23-s − 25-s + 6·26-s − 4·28-s + 6·29-s + 32-s − 2·34-s + 8·35-s − 6·37-s − 2·40-s − 6·41-s + 4·43-s + 44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s + 0.353·8-s − 0.632·10-s + 0.301·11-s + 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.447·20-s + 0.213·22-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.755·28-s + 1.11·29-s + 0.176·32-s − 0.342·34-s + 1.35·35-s − 0.986·37-s − 0.316·40-s − 0.937·41-s + 0.609·43-s + 0.150·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71478 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71478 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.14507442429536, −13.78683699094356, −13.39478180401841, −12.95184744343088, −12.30431276628231, −11.95068114895634, −11.59978954043781, −10.84923537205383, −10.41653372384358, −10.03279306231551, −9.145675398245164, −8.770029536861612, −8.298788856364332, −7.501055813261351, −7.089277266825897, −6.381840619827810, −6.160394438963956, −5.632752695144147, −4.727922018778844, −3.982508919981499, −3.846174013983410, −3.232894983803653, −2.669621861977100, −1.751135672591534, −0.8625921624658588, 0,
0.8625921624658588, 1.751135672591534, 2.669621861977100, 3.232894983803653, 3.846174013983410, 3.982508919981499, 4.727922018778844, 5.632752695144147, 6.160394438963956, 6.381840619827810, 7.089277266825897, 7.501055813261351, 8.298788856364332, 8.770029536861612, 9.145675398245164, 10.03279306231551, 10.41653372384358, 10.84923537205383, 11.59978954043781, 11.95068114895634, 12.30431276628231, 12.95184744343088, 13.39478180401841, 13.78683699094356, 14.14507442429536
