| L(s) = 1 | + 3.11i·5-s − 7-s − 2.37i·11-s − 1.09i·13-s + 3.69·17-s + 1.08i·19-s − 4.87·23-s − 4.69·25-s + 1.59i·29-s − 7.45·31-s − 3.11i·35-s + 4.61i·37-s − 0.0380·41-s − 11.0i·43-s − 0.337·47-s + ⋯ |
| L(s) = 1 | + 1.39i·5-s − 0.377·7-s − 0.715i·11-s − 0.302i·13-s + 0.895·17-s + 0.248i·19-s − 1.01·23-s − 0.939·25-s + 0.296i·29-s − 1.33·31-s − 0.526i·35-s + 0.758i·37-s − 0.00594·41-s − 1.68i·43-s − 0.0491·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0430 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0430 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7642840685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7642840685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 3.11iT - 5T^{2} \) |
| 11 | \( 1 + 2.37iT - 11T^{2} \) |
| 13 | \( 1 + 1.09iT - 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 1.08iT - 19T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 - 1.59iT - 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 - 4.61iT - 37T^{2} \) |
| 41 | \( 1 + 0.0380T + 41T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 + 0.337T + 47T^{2} \) |
| 53 | \( 1 + 8.14iT - 53T^{2} \) |
| 59 | \( 1 + 15.0iT - 59T^{2} \) |
| 61 | \( 1 + 5.53iT - 61T^{2} \) |
| 67 | \( 1 - 7.70iT - 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 - 0.138iT - 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
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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
−7.897804046003496604973218528586, −7.05578723092797499128384630747, −6.60834172732248166906354314319, −5.78193162649099072323444749886, −5.29424788686246327947955179861, −3.87010687911252628418877349182, −3.44650500469602123125135034917, −2.72947992555364940246738069246, −1.73192316738872415306734782096, −0.20471044139538925537361348673,
1.07452702839921028914304314145, 1.90201519374852987342094639195, 3.00172853691413380287350685750, 4.14426718202127354702885706702, 4.46123214668095912400075296414, 5.50834074203584264190434170269, 5.85114632776293650421224980470, 6.94045573742407206920537436356, 7.61633433911759707411404454065, 8.238611943992951266368427256317
