| L(s) = 1 | − 1.70·2-s + 3·3-s − 5.09·4-s − 1.03·5-s − 5.11·6-s + 1.91·7-s + 22.3·8-s + 9·9-s + 1.76·10-s − 11·11-s − 15.2·12-s + 24.2·13-s − 3.26·14-s − 3.09·15-s + 2.65·16-s + 33.3·17-s − 15.3·18-s + 11.9·19-s + 5.26·20-s + 5.73·21-s + 18.7·22-s − 174.·23-s + 66.9·24-s − 123.·25-s − 41.3·26-s + 27·27-s − 9.73·28-s + ⋯ |
| L(s) = 1 | − 0.602·2-s + 0.577·3-s − 0.636·4-s − 0.0924·5-s − 0.348·6-s + 0.103·7-s + 0.986·8-s + 0.333·9-s + 0.0557·10-s − 0.301·11-s − 0.367·12-s + 0.516·13-s − 0.0622·14-s − 0.0533·15-s + 0.0414·16-s + 0.476·17-s − 0.200·18-s + 0.143·19-s + 0.0588·20-s + 0.0596·21-s + 0.181·22-s − 1.57·23-s + 0.569·24-s − 0.991·25-s − 0.311·26-s + 0.192·27-s − 0.0657·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.541635522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541635522\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 1.70T + 8T^{2} \) |
| 5 | \( 1 + 1.03T + 125T^{2} \) |
| 7 | \( 1 - 1.91T + 343T^{2} \) |
| 13 | \( 1 - 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 290.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 34.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 355.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 168.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 472.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 280.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 15.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 267.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 360.T + 9.12e5T^{2} \) |
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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
−8.687058294129553556080315960622, −8.035698638482116793812028475399, −7.74669549731116538997578887115, −6.52361468678641568443243117230, −5.61597389387216059315809574252, −4.53560904058296724438013674418, −3.93547398426226643517148327092, −2.83444918176560073252005426578, −1.65900172668956633361802508727, −0.63622108154035760435343076155,
0.63622108154035760435343076155, 1.65900172668956633361802508727, 2.83444918176560073252005426578, 3.93547398426226643517148327092, 4.53560904058296724438013674418, 5.61597389387216059315809574252, 6.52361468678641568443243117230, 7.74669549731116538997578887115, 8.035698638482116793812028475399, 8.687058294129553556080315960622
