| L(s) = 1 | + 2·5-s + 7-s − 3·11-s − 3·17-s + 19-s − 6·23-s − 25-s + 5·29-s + 6·31-s + 2·35-s + 2·37-s + 41-s − 10·43-s − 3·47-s + 49-s + 11·53-s − 6·55-s + 6·59-s + 3·61-s + 2·67-s + 16·71-s − 4·73-s − 3·77-s + 11·79-s − 4·83-s − 6·85-s + 7·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.904·11-s − 0.727·17-s + 0.229·19-s − 1.25·23-s − 1/5·25-s + 0.928·29-s + 1.07·31-s + 0.338·35-s + 0.328·37-s + 0.156·41-s − 1.52·43-s − 0.437·47-s + 1/7·49-s + 1.51·53-s − 0.809·55-s + 0.781·59-s + 0.384·61-s + 0.244·67-s + 1.89·71-s − 0.468·73-s − 0.341·77-s + 1.23·79-s − 0.439·83-s − 0.650·85-s + 0.741·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543320243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543320243\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.59027966965423, −12.05899764936932, −11.70189580628879, −11.24376005431224, −10.63651269249965, −10.22280151326199, −9.925503203356199, −9.547435136155623, −8.865104552560859, −8.372943318929887, −8.062029879919458, −7.631227887564562, −6.845049220031205, −6.525209856995815, −6.076804922629412, −5.416925675001147, −5.168372182154278, −4.599650312047562, −4.030534558743639, −3.478983170307898, −2.636810392716104, −2.362184261777238, −1.872330428716005, −1.142126640506821, −0.4210854710889512,
0.4210854710889512, 1.142126640506821, 1.872330428716005, 2.362184261777238, 2.636810392716104, 3.478983170307898, 4.030534558743639, 4.599650312047562, 5.168372182154278, 5.416925675001147, 6.076804922629412, 6.525209856995815, 6.845049220031205, 7.631227887564562, 8.062029879919458, 8.372943318929887, 8.865104552560859, 9.547435136155623, 9.925503203356199, 10.22280151326199, 10.63651269249965, 11.24376005431224, 11.70189580628879, 12.05899764936932, 12.59027966965423
