| L(s) = 1 | − 1.12·2-s − 0.740·4-s − 1.11·7-s + 3.07·8-s − 3.67·11-s + 4.05·13-s + 1.24·14-s − 1.97·16-s + 2.12·17-s − 4.06·19-s + 4.11·22-s + 6.17·23-s − 4.54·26-s + 0.824·28-s + 2.25·29-s + 10.0·31-s − 3.93·32-s − 2.38·34-s − 7.37·37-s + 4.55·38-s + 7.47·41-s − 9.24·43-s + 2.71·44-s − 6.92·46-s − 3.12·47-s − 5.75·49-s − 3.00·52-s + ⋯ |
| L(s) = 1 | − 0.793·2-s − 0.370·4-s − 0.420·7-s + 1.08·8-s − 1.10·11-s + 1.12·13-s + 0.334·14-s − 0.492·16-s + 0.514·17-s − 0.931·19-s + 0.878·22-s + 1.28·23-s − 0.891·26-s + 0.155·28-s + 0.419·29-s + 1.80·31-s − 0.696·32-s − 0.408·34-s − 1.21·37-s + 0.739·38-s + 1.16·41-s − 1.41·43-s + 0.409·44-s − 1.02·46-s − 0.456·47-s − 0.822·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8670386142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8670386142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 7.37T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 + 0.660T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.281280466377163755323122032537, −7.69571743813020878252262186135, −6.77082954861717584593815154315, −6.15103489562125533885811920476, −5.12978122329289916083181586805, −4.61438942208781553252984715610, −3.57365178332189312711477260012, −2.80911642572304751321719322983, −1.58116148754028389986090257311, −0.58683403211265121354573265555,
0.58683403211265121354573265555, 1.58116148754028389986090257311, 2.80911642572304751321719322983, 3.57365178332189312711477260012, 4.61438942208781553252984715610, 5.12978122329289916083181586805, 6.15103489562125533885811920476, 6.77082954861717584593815154315, 7.69571743813020878252262186135, 8.281280466377163755323122032537
