| L(s) = 1 | + 3-s + 9-s + 3·11-s + 17-s + 2·19-s + 23-s + 27-s + 6·29-s + 5·31-s + 3·33-s − 4·37-s + 8·41-s − 7·47-s + 51-s + 6·53-s + 2·57-s + 5·59-s − 13·61-s − 5·67-s + 69-s − 5·71-s + 3·73-s − 8·79-s + 81-s − 83-s + 6·87-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.904·11-s + 0.242·17-s + 0.458·19-s + 0.208·23-s + 0.192·27-s + 1.11·29-s + 0.898·31-s + 0.522·33-s − 0.657·37-s + 1.24·41-s − 1.02·47-s + 0.140·51-s + 0.824·53-s + 0.264·57-s + 0.650·59-s − 1.66·61-s − 0.610·67-s + 0.120·69-s − 0.593·71-s + 0.351·73-s − 0.900·79-s + 1/9·81-s − 0.109·83-s + 0.643·87-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.18195430864636, −12.57862677761912, −12.08906642547445, −11.85209071220834, −11.28935626906398, −10.70190891502391, −10.24292235473470, −9.807944631005562, −9.329402469713204, −8.856623940141281, −8.526015249632231, −7.883277307032640, −7.534700887579731, −6.944062243972022, −6.435125504193824, −6.111796211850324, −5.320454968874463, −4.890692379406393, −4.207267253511853, −3.922237364138211, −3.134989483022320, −2.825362834164725, −2.162946219098628, −1.282551660648206, −1.102729054067335, 0,
1.102729054067335, 1.282551660648206, 2.162946219098628, 2.825362834164725, 3.134989483022320, 3.922237364138211, 4.207267253511853, 4.890692379406393, 5.320454968874463, 6.111796211850324, 6.435125504193824, 6.944062243972022, 7.534700887579731, 7.883277307032640, 8.526015249632231, 8.856623940141281, 9.329402469713204, 9.807944631005562, 10.24292235473470, 10.70190891502391, 11.28935626906398, 11.85209071220834, 12.08906642547445, 12.57862677761912, 13.18195430864636
