| L(s) = 1 | − 0.297·2-s − 1.91·4-s + 3.06·5-s + 7-s + 1.16·8-s − 0.911·10-s + 5.38·13-s − 0.297·14-s + 3.47·16-s − 6.72·17-s − 0.434·19-s − 5.85·20-s − 2.92·23-s + 4.38·25-s − 1.60·26-s − 1.91·28-s + 6.86·29-s + 4.95·31-s − 3.36·32-s + 2.00·34-s + 3.06·35-s − 3.38·37-s + 0.129·38-s + 3.56·40-s + 9.92·41-s − 1.82·43-s + 0.869·46-s + ⋯ |
| L(s) = 1 | − 0.210·2-s − 0.955·4-s + 1.37·5-s + 0.377·7-s + 0.411·8-s − 0.288·10-s + 1.49·13-s − 0.0795·14-s + 0.869·16-s − 1.63·17-s − 0.0997·19-s − 1.30·20-s − 0.609·23-s + 0.877·25-s − 0.314·26-s − 0.361·28-s + 1.27·29-s + 0.889·31-s − 0.594·32-s + 0.342·34-s + 0.517·35-s − 0.557·37-s + 0.0209·38-s + 0.563·40-s + 1.55·41-s − 0.278·43-s + 0.128·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.136228545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.136228545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.297T + 2T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 + 0.434T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 31 | \( 1 - 4.95T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 - 9.33T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 6.77T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 8.50T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 6.95T + 97T^{2} \) |
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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
−8.249924968132566865940693525878, −7.12881061900871947752096202142, −6.17919115593763189396921990154, −6.01409944108048749592827996101, −4.97829600757584506667616981428, −4.45736594308547414462347625468, −3.64818159727377542555943145031, −2.50275006919990130670322368239, −1.69313217360840606079657056050, −0.802716779898837679115979525407,
0.802716779898837679115979525407, 1.69313217360840606079657056050, 2.50275006919990130670322368239, 3.64818159727377542555943145031, 4.45736594308547414462347625468, 4.97829600757584506667616981428, 6.01409944108048749592827996101, 6.17919115593763189396921990154, 7.12881061900871947752096202142, 8.249924968132566865940693525878
