| L(s) = 1 | − 1.26·2-s − 1.64·3-s − 0.397·4-s − 2.39·5-s + 2.08·6-s − 1.52·7-s + 3.03·8-s − 0.297·9-s + 3.03·10-s − 1.91·11-s + 0.653·12-s + 13-s + 1.92·14-s + 3.93·15-s − 3.04·16-s + 4.22·17-s + 0.376·18-s + 6.85·19-s + 0.951·20-s + 2.50·21-s + 2.42·22-s − 3.30·23-s − 4.98·24-s + 0.734·25-s − 1.26·26-s + 5.42·27-s + 0.605·28-s + ⋯ |
| L(s) = 1 | − 0.895·2-s − 0.949·3-s − 0.198·4-s − 1.07·5-s + 0.849·6-s − 0.575·7-s + 1.07·8-s − 0.0992·9-s + 0.958·10-s − 0.577·11-s + 0.188·12-s + 0.277·13-s + 0.515·14-s + 1.01·15-s − 0.761·16-s + 1.02·17-s + 0.0888·18-s + 1.57·19-s + 0.212·20-s + 0.546·21-s + 0.516·22-s − 0.689·23-s − 1.01·24-s + 0.146·25-s − 0.248·26-s + 1.04·27-s + 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 - 6.83T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.84T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.754T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 4.11T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 0.414T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.346791553827979718956412439128, −8.236503229532030895176199195858, −7.79331463749938515348594056371, −6.98591663920753732513087675332, −5.82174990367558363483321000041, −5.11257586325830831376764064713, −4.05250764196380133463671658225, −3.04754534909326483033127848357, −1.02876837709624999313364571227, 0,
1.02876837709624999313364571227, 3.04754534909326483033127848357, 4.05250764196380133463671658225, 5.11257586325830831376764064713, 5.82174990367558363483321000041, 6.98591663920753732513087675332, 7.79331463749938515348594056371, 8.236503229532030895176199195858, 9.346791553827979718956412439128
