| L(s) = 1 | + 0.783·3-s + 3.56·5-s + 7-s − 2.38·9-s + 2.51·11-s + 6.52·13-s + 2.79·15-s − 1.93·17-s + 6.56·19-s + 0.783·21-s + 2.76·23-s + 7.73·25-s − 4.21·27-s + 2.34·29-s − 7.50·31-s + 1.97·33-s + 3.56·35-s + 6.05·37-s + 5.11·39-s − 7.71·41-s − 10.9·43-s − 8.51·45-s − 8.18·47-s + 49-s − 1.51·51-s − 2.04·53-s + 8.98·55-s + ⋯ |
| L(s) = 1 | + 0.452·3-s + 1.59·5-s + 0.377·7-s − 0.795·9-s + 0.759·11-s + 1.80·13-s + 0.721·15-s − 0.470·17-s + 1.50·19-s + 0.170·21-s + 0.577·23-s + 1.54·25-s − 0.811·27-s + 0.435·29-s − 1.34·31-s + 0.343·33-s + 0.603·35-s + 0.994·37-s + 0.818·39-s − 1.20·41-s − 1.66·43-s − 1.26·45-s − 1.19·47-s + 0.142·49-s − 0.212·51-s − 0.281·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.542025306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.542025306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 0.783T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 - 6.52T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 7.71T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 8.18T + 47T^{2} \) |
| 53 | \( 1 + 2.04T + 53T^{2} \) |
| 59 | \( 1 - 0.911T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.689293411000787059561730045778, −8.062247175130144355830775158113, −6.87036135067481996252336337113, −6.27723286174869977234165694029, −5.60467794577375607102852696982, −4.98117112305303506143320564593, −3.64822398708077025686890462434, −3.02717860626917723437702259344, −1.87243871117039861411270413740, −1.24094581391924270128415359521,
1.24094581391924270128415359521, 1.87243871117039861411270413740, 3.02717860626917723437702259344, 3.64822398708077025686890462434, 4.98117112305303506143320564593, 5.60467794577375607102852696982, 6.27723286174869977234165694029, 6.87036135067481996252336337113, 8.062247175130144355830775158113, 8.689293411000787059561730045778
