| L(s) = 1 | − 0.462·5-s − 0.323·7-s − 5.58·11-s + 6.58·13-s + 6.64·17-s − 2.64·19-s − 4.86·23-s − 4.78·25-s + 4.86·29-s + 6.46·31-s + 0.149·35-s − 37-s + 0.815·41-s − 1.87·43-s + 1.11·47-s − 6.89·49-s + 12.6·53-s + 2.58·55-s + 0.128·59-s − 1.10·61-s − 3.04·65-s + 13.4·67-s + 8.04·71-s + 3.58·73-s + 1.80·77-s − 3.78·79-s + 14.6·83-s + ⋯ |
| L(s) = 1 | − 0.206·5-s − 0.122·7-s − 1.68·11-s + 1.82·13-s + 1.61·17-s − 0.607·19-s − 1.01·23-s − 0.957·25-s + 0.902·29-s + 1.16·31-s + 0.0252·35-s − 0.164·37-s + 0.127·41-s − 0.285·43-s + 0.163·47-s − 0.985·49-s + 1.74·53-s + 0.348·55-s + 0.0167·59-s − 0.142·61-s − 0.377·65-s + 1.64·67-s + 0.954·71-s + 0.419·73-s + 0.205·77-s − 0.425·79-s + 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.619073396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.619073396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 0.462T + 5T^{2} \) |
| 7 | \( 1 + 0.323T + 7T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 41 | \( 1 - 0.815T + 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 0.128T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 8.04T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−8.493213330634301992391620167708, −8.205223826575047887440897303239, −7.55776317970792062193348957069, −6.38189968620705610356306366452, −5.82742837630452335925619043608, −5.02991098804043194414009689911, −3.93216035221722942357898506867, −3.23673672930971092103283349476, −2.15788028160567918688415226779, −0.798674056178037082436618027808,
0.798674056178037082436618027808, 2.15788028160567918688415226779, 3.23673672930971092103283349476, 3.93216035221722942357898506867, 5.02991098804043194414009689911, 5.82742837630452335925619043608, 6.38189968620705610356306366452, 7.55776317970792062193348957069, 8.205223826575047887440897303239, 8.493213330634301992391620167708
