| L(s) = 1 | + 0.171·2-s − 3-s − 1.97·4-s − 0.133·5-s − 0.171·6-s + 0.615·7-s − 0.681·8-s + 9-s − 0.0228·10-s + 11-s + 1.97·12-s − 4.70·13-s + 0.105·14-s + 0.133·15-s + 3.82·16-s + 2.34·17-s + 0.171·18-s − 7.85·19-s + 0.262·20-s − 0.615·21-s + 0.171·22-s + 1.86·23-s + 0.681·24-s − 4.98·25-s − 0.807·26-s − 27-s − 1.21·28-s + ⋯ |
| L(s) = 1 | + 0.121·2-s − 0.577·3-s − 0.985·4-s − 0.0595·5-s − 0.0700·6-s + 0.232·7-s − 0.240·8-s + 0.333·9-s − 0.00722·10-s + 0.301·11-s + 0.568·12-s − 1.30·13-s + 0.0282·14-s + 0.0343·15-s + 0.956·16-s + 0.568·17-s + 0.0404·18-s − 1.80·19-s + 0.0586·20-s − 0.134·21-s + 0.0365·22-s + 0.389·23-s + 0.139·24-s − 0.996·25-s − 0.158·26-s − 0.192·27-s − 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8978192655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8978192655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.171T + 2T^{2} \) |
| 5 | \( 1 + 0.133T + 5T^{2} \) |
| 7 | \( 1 - 0.615T + 7T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 - 4.53T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 9.24T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 7.78T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.320853990356033054375347454528, −8.247970975570384995478355415080, −7.79111564280861012683442730199, −6.60887773250508544620429449470, −5.96140346536296872716412292374, −4.83075774965265319389544011159, −4.58638117128779470336618925233, −3.50114109753365180589810462198, −2.16074546505353917118139158719, −0.63253096697509489727527879514,
0.63253096697509489727527879514, 2.16074546505353917118139158719, 3.50114109753365180589810462198, 4.58638117128779470336618925233, 4.83075774965265319389544011159, 5.96140346536296872716412292374, 6.60887773250508544620429449470, 7.79111564280861012683442730199, 8.247970975570384995478355415080, 9.320853990356033054375347454528
