| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 3·11-s − 15-s − 3·17-s − 7·19-s − 21-s − 3·23-s + 25-s + 5·27-s − 3·29-s + 4·31-s − 3·33-s + 35-s − 7·37-s + 9·41-s − 11·43-s − 2·45-s − 6·49-s + 3·51-s + 6·53-s + 3·55-s + 7·57-s − 3·59-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.258·15-s − 0.727·17-s − 1.60·19-s − 0.218·21-s − 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.718·31-s − 0.522·33-s + 0.169·35-s − 1.15·37-s + 1.40·41-s − 1.67·43-s − 0.298·45-s − 6/7·49-s + 0.420·51-s + 0.824·53-s + 0.404·55-s + 0.927·57-s − 0.390·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9853818981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9853818981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−14.33970098333280, −14.02331087400111, −13.43857750403240, −12.87478383339794, −12.29072070450517, −11.88969955858104, −11.24787571717119, −11.02250900006252, −10.33346590580732, −9.943762353064706, −9.031728177179526, −8.839538925825915, −8.278775100445448, −7.633409088800115, −6.745109327149849, −6.470060015713274, −5.975190930047349, −5.409474979599959, −4.647089536221756, −4.305943791479944, −3.509621986223500, −2.717019973229170, −1.985245983753143, −1.463525478288687, −0.3481098279981124,
0.3481098279981124, 1.463525478288687, 1.985245983753143, 2.717019973229170, 3.509621986223500, 4.305943791479944, 4.647089536221756, 5.409474979599959, 5.975190930047349, 6.470060015713274, 6.745109327149849, 7.633409088800115, 8.278775100445448, 8.839538925825915, 9.031728177179526, 9.943762353064706, 10.33346590580732, 11.02250900006252, 11.24787571717119, 11.88969955858104, 12.29072070450517, 12.87478383339794, 13.43857750403240, 14.02331087400111, 14.33970098333280
