| L(s) = 1 | + 3.64·5-s + 4.81·7-s − 2.68·11-s − 3.85·13-s + 0.962·17-s − 4.21·19-s + 23-s + 8.29·25-s + 7.85·29-s − 7.85·31-s + 17.5·35-s + 11.0·37-s + 0.564·41-s + 5.57·43-s − 7.63·47-s + 16.2·49-s + 9.99·53-s − 9.78·55-s + 15.0·59-s − 0.683·61-s − 14.0·65-s + 10.7·67-s + 3.43·71-s + 0.0692·73-s − 12.9·77-s − 6.89·79-s + 1.31·83-s + ⋯ |
| L(s) = 1 | + 1.63·5-s + 1.82·7-s − 0.808·11-s − 1.06·13-s + 0.233·17-s − 0.965·19-s + 0.208·23-s + 1.65·25-s + 1.45·29-s − 1.41·31-s + 2.96·35-s + 1.81·37-s + 0.0881·41-s + 0.850·43-s − 1.11·47-s + 2.31·49-s + 1.37·53-s − 1.31·55-s + 1.96·59-s − 0.0874·61-s − 1.74·65-s + 1.31·67-s + 0.407·71-s + 0.00809·73-s − 1.47·77-s − 0.775·79-s + 0.144·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.044141284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.044141284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 - 4.81T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 0.962T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + 7.85T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 0.564T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 - 9.99T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 0.683T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 - 0.0692T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 - 1.31T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 + 4.34T + 97T^{2} \) |
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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
−8.575130633473291541202372990911, −7.972171767835657039493701540414, −7.19815154727876742856517265509, −6.28015954078148862630983594066, −5.32979695609213789300019574835, −5.10991376729094436419768349119, −4.21392792790787580774487762792, −2.45034783982930936247893157199, −2.25864396690449414035366064835, −1.11685231277491393902089433064,
1.11685231277491393902089433064, 2.25864396690449414035366064835, 2.45034783982930936247893157199, 4.21392792790787580774487762792, 5.10991376729094436419768349119, 5.32979695609213789300019574835, 6.28015954078148862630983594066, 7.19815154727876742856517265509, 7.972171767835657039493701540414, 8.575130633473291541202372990911
