| L(s) = 1 | + 4.18·5-s + 0.147·7-s − 1.80·11-s + 2.10·13-s + 5.26·17-s − 7.39·19-s + 4.13·23-s + 12.5·25-s − 5.71·29-s + 6.06·31-s + 0.617·35-s + 7.57·37-s + 1.89·41-s + 2.31·43-s + 7.95·47-s − 6.97·49-s − 2.04·53-s − 7.53·55-s − 14.5·59-s + 3.39·61-s + 8.81·65-s + 11.7·67-s − 7.41·71-s − 11.3·73-s − 0.265·77-s + 15.3·79-s − 1.41·83-s + ⋯ |
| L(s) = 1 | + 1.87·5-s + 0.0557·7-s − 0.543·11-s + 0.584·13-s + 1.27·17-s − 1.69·19-s + 0.862·23-s + 2.50·25-s − 1.06·29-s + 1.08·31-s + 0.104·35-s + 1.24·37-s + 0.295·41-s + 0.353·43-s + 1.16·47-s − 0.996·49-s − 0.280·53-s − 1.01·55-s − 1.89·59-s + 0.435·61-s + 1.09·65-s + 1.43·67-s − 0.879·71-s − 1.33·73-s − 0.0302·77-s + 1.73·79-s − 0.155·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599604821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599604821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 - 0.147T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 - 5.26T + 17T^{2} \) |
| 19 | \( 1 + 7.39T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 + 5.71T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 - 7.57T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 + 2.04T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 7.41T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.308679604833224376471141903824, −8.544934315575885722673639824879, −7.66674870911193406614304454082, −6.54890069768603774199610190670, −5.99549898530239770974637685174, −5.36718759216377998904291956379, −4.41523474874225932362383132002, −3.04011578963157952206065252173, −2.20852802038008707569279967401, −1.18747686280476757537969962770,
1.18747686280476757537969962770, 2.20852802038008707569279967401, 3.04011578963157952206065252173, 4.41523474874225932362383132002, 5.36718759216377998904291956379, 5.99549898530239770974637685174, 6.54890069768603774199610190670, 7.66674870911193406614304454082, 8.544934315575885722673639824879, 9.308679604833224376471141903824
