| L(s) = 1 | + 3.95·5-s − 2.34·7-s − 11-s − 13-s − 7.66·17-s − 1.89·19-s + 6.65·23-s + 10.6·25-s − 2.03·29-s + 5.72·31-s − 9.27·35-s − 0.448·37-s − 7.75·41-s + 10.1·43-s + 7.14·47-s − 1.49·49-s + 4.44·53-s − 3.95·55-s + 14.6·59-s + 11.5·61-s − 3.95·65-s − 2.87·67-s − 2.35·71-s + 15.4·73-s + 2.34·77-s + 8.11·79-s + 13.2·83-s + ⋯ |
| L(s) = 1 | + 1.76·5-s − 0.886·7-s − 0.301·11-s − 0.277·13-s − 1.85·17-s − 0.435·19-s + 1.38·23-s + 2.12·25-s − 0.377·29-s + 1.02·31-s − 1.56·35-s − 0.0737·37-s − 1.21·41-s + 1.54·43-s + 1.04·47-s − 0.213·49-s + 0.611·53-s − 0.532·55-s + 1.90·59-s + 1.47·61-s − 0.490·65-s − 0.350·67-s − 0.279·71-s + 1.80·73-s + 0.267·77-s + 0.913·79-s + 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299512892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299512892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.95T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 17 | \( 1 + 7.66T + 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 + 0.448T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 8.11T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 1.34T + 97T^{2} \) |
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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
−8.499749399483841070095719425435, −7.21513588125313849319260740711, −6.59459403410083549637020948852, −6.23675893284177977428896294089, −5.31491664976254434900473506561, −4.76611195160117177126265105412, −3.62760500672824000976206555812, −2.41762870350303489534963867684, −2.28953589915151685540197267597, −0.806127385651519224847186108588,
0.806127385651519224847186108588, 2.28953589915151685540197267597, 2.41762870350303489534963867684, 3.62760500672824000976206555812, 4.76611195160117177126265105412, 5.31491664976254434900473506561, 6.23675893284177977428896294089, 6.59459403410083549637020948852, 7.21513588125313849319260740711, 8.499749399483841070095719425435
