| L(s) = 1 | − 1.36·2-s − 3-s − 0.149·4-s + 1.36·6-s + 2.92·8-s + 9-s + 0.625·11-s + 0.149·12-s − 0.211·13-s − 3.67·16-s + 0.904·17-s − 1.36·18-s − 0.808·19-s − 0.850·22-s − 5.75·23-s − 2.92·24-s + 0.287·26-s − 27-s + 3.22·29-s + 9.68·31-s − 0.842·32-s − 0.625·33-s − 1.23·34-s − 0.149·36-s − 2.89·37-s + 1.09·38-s + 0.211·39-s + ⋯ |
| L(s) = 1 | − 0.961·2-s − 0.577·3-s − 0.0746·4-s + 0.555·6-s + 1.03·8-s + 0.333·9-s + 0.188·11-s + 0.0430·12-s − 0.0585·13-s − 0.919·16-s + 0.219·17-s − 0.320·18-s − 0.185·19-s − 0.181·22-s − 1.19·23-s − 0.596·24-s + 0.0563·26-s − 0.192·27-s + 0.598·29-s + 1.73·31-s − 0.148·32-s − 0.108·33-s − 0.211·34-s − 0.0248·36-s − 0.475·37-s + 0.178·38-s + 0.0338·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7064712176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7064712176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 11 | \( 1 - 0.625T + 11T^{2} \) |
| 13 | \( 1 + 0.211T + 13T^{2} \) |
| 17 | \( 1 - 0.904T + 17T^{2} \) |
| 19 | \( 1 + 0.808T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 - 9.68T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 + 1.36T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.602108764718686197868967874292, −7.82586767535966381109470167588, −7.30604061084399598614885252421, −6.30397049913267086520345299644, −5.72219015220486822759743980260, −4.54443084689273328573776261731, −4.21735874689298236823742854914, −2.80004669064491193033732070589, −1.59911772194571610657056308322, −0.61656827431226732506752616010,
0.61656827431226732506752616010, 1.59911772194571610657056308322, 2.80004669064491193033732070589, 4.21735874689298236823742854914, 4.54443084689273328573776261731, 5.72219015220486822759743980260, 6.30397049913267086520345299644, 7.30604061084399598614885252421, 7.82586767535966381109470167588, 8.602108764718686197868967874292
