| L(s) = 1 | + 3-s − 2·5-s − 2·9-s − 2·11-s + 7·13-s − 2·15-s + 4·17-s + 6·19-s + 23-s − 25-s − 5·27-s − 5·29-s + 3·31-s − 2·33-s − 2·37-s + 7·39-s + 9·41-s + 8·43-s + 4·45-s − 47-s + 4·51-s + 6·53-s + 4·55-s + 6·57-s + 8·59-s − 10·61-s − 14·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 2/3·9-s − 0.603·11-s + 1.94·13-s − 0.516·15-s + 0.970·17-s + 1.37·19-s + 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s + 0.538·31-s − 0.348·33-s − 0.328·37-s + 1.12·39-s + 1.40·41-s + 1.21·43-s + 0.596·45-s − 0.145·47-s + 0.560·51-s + 0.824·53-s + 0.539·55-s + 0.794·57-s + 1.04·59-s − 1.28·61-s − 1.73·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.845969901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.845969901\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.04885477200229, −13.66853895725332, −13.25955998871503, −12.60230411062998, −12.08470234343311, −11.45496418943378, −11.23942434049105, −10.71610971347495, −10.07315602390811, −9.331056108301278, −9.070130861574149, −8.317563221494067, −8.027684616247215, −7.602247941012858, −7.098146576646708, −6.169915713740024, −5.692333034742318, −5.367780385843659, −4.409910069312911, −3.718930919329924, −3.473673060207226, −2.901042030555378, −2.147270684805578, −1.164274624940970, −0.6115578375487985,
0.6115578375487985, 1.164274624940970, 2.147270684805578, 2.901042030555378, 3.473673060207226, 3.718930919329924, 4.409910069312911, 5.367780385843659, 5.692333034742318, 6.169915713740024, 7.098146576646708, 7.602247941012858, 8.027684616247215, 8.317563221494067, 9.070130861574149, 9.331056108301278, 10.07315602390811, 10.71610971347495, 11.23942434049105, 11.45496418943378, 12.08470234343311, 12.60230411062998, 13.25955998871503, 13.66853895725332, 14.04885477200229
