| L(s) = 1 | − 3·5-s + 3·11-s + 13-s − 3·17-s − 7·19-s + 9·23-s + 4·25-s + 3·29-s + 8·31-s − 37-s − 3·41-s + 43-s + 3·53-s − 9·55-s − 2·61-s − 3·65-s + 4·67-s − 12·71-s − 11·73-s + 16·79-s − 9·83-s + 9·85-s − 3·89-s + 21·95-s + 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.60·19-s + 1.87·23-s + 4/5·25-s + 0.557·29-s + 1.43·31-s − 0.164·37-s − 0.468·41-s + 0.152·43-s + 0.412·53-s − 1.21·55-s − 0.256·61-s − 0.372·65-s + 0.488·67-s − 1.42·71-s − 1.28·73-s + 1.80·79-s − 0.987·83-s + 0.976·85-s − 0.317·89-s + 2.15·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−14.67469094802503, −14.05322516919429, −13.30099494220624, −13.11365372909369, −12.29535966316467, −12.01059460161414, −11.50814445505049, −10.93355590313593, −10.69108294192180, −9.979339180594490, −9.153412389878118, −8.808173934885900, −8.357206621500006, −7.894610839760274, −7.097746801870335, −6.695062108046194, −6.383813351686505, −5.473538149195049, −4.669423304225412, −4.317045302211745, −3.889035420323941, −3.096528942937501, −2.593400386942858, −1.577081974945528, −0.8486786059420609, 0,
0.8486786059420609, 1.577081974945528, 2.593400386942858, 3.096528942937501, 3.889035420323941, 4.317045302211745, 4.669423304225412, 5.473538149195049, 6.383813351686505, 6.695062108046194, 7.097746801870335, 7.894610839760274, 8.357206621500006, 8.808173934885900, 9.153412389878118, 9.979339180594490, 10.69108294192180, 10.93355590313593, 11.50814445505049, 12.01059460161414, 12.29535966316467, 13.11365372909369, 13.30099494220624, 14.05322516919429, 14.67469094802503
