| L(s) = 1 | + 0.849·2-s − 1.27·4-s − 0.881·5-s + 3.72·7-s − 2.78·8-s − 0.749·10-s − 4.72·11-s + 4.38·13-s + 3.16·14-s + 0.189·16-s + 4.69·17-s + 6.60·19-s + 1.12·20-s − 4.01·22-s − 3.92·23-s − 4.22·25-s + 3.72·26-s − 4.75·28-s − 6.53·29-s + 5.73·32-s + 3.99·34-s − 3.28·35-s + 5.55·37-s + 5.61·38-s + 2.45·40-s − 1.23·41-s + 0.461·43-s + ⋯ |
| L(s) = 1 | + 0.600·2-s − 0.639·4-s − 0.394·5-s + 1.40·7-s − 0.984·8-s − 0.236·10-s − 1.42·11-s + 1.21·13-s + 0.845·14-s + 0.0474·16-s + 1.13·17-s + 1.51·19-s + 0.251·20-s − 0.855·22-s − 0.818·23-s − 0.844·25-s + 0.730·26-s − 0.898·28-s − 1.21·29-s + 1.01·32-s + 0.684·34-s − 0.554·35-s + 0.913·37-s + 0.910·38-s + 0.388·40-s − 0.192·41-s + 0.0704·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328891991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328891991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.849T + 2T^{2} \) |
| 5 | \( 1 + 0.881T + 5T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 37 | \( 1 - 5.55T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 - 0.461T + 43T^{2} \) |
| 47 | \( 1 - 1.88T + 47T^{2} \) |
| 53 | \( 1 - 8.04T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 + 0.565T + 71T^{2} \) |
| 73 | \( 1 + 6.64T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−7.72190209414429574532430778054, −7.47099951887715798599032398252, −5.89412992595754982380684207839, −5.62493867702683299889002213480, −5.05846428121311150806852726641, −4.24255177523517299835207347926, −3.64887986819829631774465979483, −2.88860836986414265853577702850, −1.72632486366405347196998767054, −0.70524698984191975753322985625,
0.70524698984191975753322985625, 1.72632486366405347196998767054, 2.88860836986414265853577702850, 3.64887986819829631774465979483, 4.24255177523517299835207347926, 5.05846428121311150806852726641, 5.62493867702683299889002213480, 5.89412992595754982380684207839, 7.47099951887715798599032398252, 7.72190209414429574532430778054
