| L(s) = 1 | + 3·5-s + 7-s + 3·11-s − 13-s − 17-s + 3·19-s − 5·23-s + 4·25-s + 3·29-s − 8·31-s + 3·35-s − 5·37-s − 12·41-s − 7·43-s − 8·47-s + 49-s + 2·53-s + 9·55-s + 61-s − 3·65-s + 14·67-s + 4·71-s − 11·73-s + 3·77-s + 6·79-s − 6·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.904·11-s − 0.277·13-s − 0.242·17-s + 0.688·19-s − 1.04·23-s + 4/5·25-s + 0.557·29-s − 1.43·31-s + 0.507·35-s − 0.821·37-s − 1.87·41-s − 1.06·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.21·55-s + 0.128·61-s − 0.372·65-s + 1.71·67-s + 0.474·71-s − 1.28·73-s + 0.341·77-s + 0.675·79-s − 0.658·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.46118356292965, −14.99019068912532, −14.24142196351809, −14.09949523712592, −13.56517531246135, −12.98068333404953, −12.37577580379847, −11.72917614176234, −11.40225124931433, −10.55866865451597, −10.00410691427375, −9.699219860184467, −9.073092523911565, −8.495717756083295, −7.941800040902678, −6.955464904480766, −6.719002094155561, −6.000235123636803, −5.298197243473604, −5.029298640354046, −4.016060731339623, −3.434019966979620, −2.519593391923669, −1.720545822273732, −1.436260233415662, 0,
1.436260233415662, 1.720545822273732, 2.519593391923669, 3.434019966979620, 4.016060731339623, 5.029298640354046, 5.298197243473604, 6.000235123636803, 6.719002094155561, 6.955464904480766, 7.941800040902678, 8.495717756083295, 9.073092523911565, 9.699219860184467, 10.00410691427375, 10.55866865451597, 11.40225124931433, 11.72917614176234, 12.37577580379847, 12.98068333404953, 13.56517531246135, 14.09949523712592, 14.24142196351809, 14.99019068912532, 15.46118356292965
