| L(s) = 1 | − 1.96·5-s + 7-s − 3.38·11-s + 6.48·13-s + 5.06·17-s − 4.13·19-s − 7.09·23-s − 1.13·25-s − 6.48·29-s + 6.34·31-s − 1.96·35-s − 6.65·37-s + 8.44·41-s + 5.65·43-s − 3.71·47-s + 49-s − 1.17·53-s + 6.65·55-s − 10.5·59-s + 10.1·61-s − 12.7·65-s + 0.216·67-s + 8.16·71-s − 12.3·73-s − 3.38·77-s − 11.3·79-s + 2.48·83-s + ⋯ |
| L(s) = 1 | − 0.879·5-s + 0.377·7-s − 1.01·11-s + 1.79·13-s + 1.22·17-s − 0.948·19-s − 1.48·23-s − 0.226·25-s − 1.20·29-s + 1.14·31-s − 0.332·35-s − 1.09·37-s + 1.31·41-s + 0.861·43-s − 0.542·47-s + 0.142·49-s − 0.160·53-s + 0.896·55-s − 1.37·59-s + 1.29·61-s − 1.58·65-s + 0.0263·67-s + 0.968·71-s − 1.44·73-s − 0.385·77-s − 1.27·79-s + 0.272·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.96T + 5T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 4.13T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 - 8.44T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 0.216T + 67T^{2} \) |
| 71 | \( 1 - 8.16T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 7.86T + 97T^{2} \) |
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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
−7.892302724430374292362758071941, −7.23549633618844797838946055653, −6.00485720056768267521795581375, −5.84022284715977683623071890315, −4.70821623877111662651496583957, −3.92907537113808629967662038523, −3.45338333184067653435572420290, −2.31559528601382538606644926080, −1.26754755950454492710176783200, 0,
1.26754755950454492710176783200, 2.31559528601382538606644926080, 3.45338333184067653435572420290, 3.92907537113808629967662038523, 4.70821623877111662651496583957, 5.84022284715977683623071890315, 6.00485720056768267521795581375, 7.23549633618844797838946055653, 7.892302724430374292362758071941
