| L(s) = 1 | + 2-s + 0.680·3-s + 4-s − 3.85·5-s + 0.680·6-s − 5.17·7-s + 8-s − 2.53·9-s − 3.85·10-s − 3.40·11-s + 0.680·12-s − 0.892·13-s − 5.17·14-s − 2.62·15-s + 16-s − 6.67·17-s − 2.53·18-s − 3.11·19-s − 3.85·20-s − 3.51·21-s − 3.40·22-s + 1.52·23-s + 0.680·24-s + 9.89·25-s − 0.892·26-s − 3.76·27-s − 5.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.392·3-s + 0.5·4-s − 1.72·5-s + 0.277·6-s − 1.95·7-s + 0.353·8-s − 0.845·9-s − 1.22·10-s − 1.02·11-s + 0.196·12-s − 0.247·13-s − 1.38·14-s − 0.678·15-s + 0.250·16-s − 1.62·17-s − 0.597·18-s − 0.715·19-s − 0.863·20-s − 0.767·21-s − 0.726·22-s + 0.318·23-s + 0.138·24-s + 1.97·25-s − 0.175·26-s − 0.725·27-s − 0.977·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1578854478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1578854478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 0.680T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 + 5.17T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 + 0.892T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 1.66T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 6.06T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 - 8.67T + 89T^{2} \) |
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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
−8.055625992482101107318259415635, −7.09551717443320311869284031600, −6.88867975568314040501238577842, −5.95319739681374212840983572930, −5.13648396773149610683769674946, −4.11675046263230600375356340205, −3.72129707192583318912333478888, −2.87390857209464358746447591964, −2.49005050894104869656296330822, −0.16480524096471953989664701544,
0.16480524096471953989664701544, 2.49005050894104869656296330822, 2.87390857209464358746447591964, 3.72129707192583318912333478888, 4.11675046263230600375356340205, 5.13648396773149610683769674946, 5.95319739681374212840983572930, 6.88867975568314040501238577842, 7.09551717443320311869284031600, 8.055625992482101107318259415635
