| L(s) = 1 | + 3-s + 3.46i·7-s − 2·9-s − 3i·11-s − 3.46·13-s + 3i·17-s + i·19-s + 3.46i·21-s − 5·27-s + 10.3i·29-s − 6.92·31-s − 3i·33-s + 10.3·37-s − 3.46·39-s − 9·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30i·7-s − 0.666·9-s − 0.904i·11-s − 0.960·13-s + 0.727i·17-s + 0.229i·19-s + 0.755i·21-s − 0.962·27-s + 1.92i·29-s − 1.24·31-s − 0.522i·33-s + 1.70·37-s − 0.554·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9513523045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9513523045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.369302902924910573684431629216, −8.931182422240695294083392412026, −8.286643298652212349461860470583, −7.53734451661307048351629413944, −6.31790909148739667868737525924, −5.66659117529891332095836629974, −4.89976401236657384114025377227, −3.45487455866185553177947095519, −2.83579584084404623751180765951, −1.81290984275541524053247721743,
0.31158091186436398135864815359, 2.00818428692560639538080067072, 2.96174008167483268024436808646, 4.05535092209468176653886286985, 4.74866849081609263614755408604, 5.82350585779589843475681879449, 7.04982931832185222436333292199, 7.42317722913850596792055557466, 8.198811359750889620586257651534, 9.220199166375498143874167652985
