L(s) = 1 | + 2-s − 2.83·3-s + 4-s + 1.28·5-s − 2.83·6-s − 4.11·7-s + 8-s + 5.01·9-s + 1.28·10-s − 5.23·11-s − 2.83·12-s + 5.59·13-s − 4.11·14-s − 3.63·15-s + 16-s + 0.524·17-s + 5.01·18-s − 19-s + 1.28·20-s + 11.6·21-s − 5.23·22-s + 5.73·23-s − 2.83·24-s − 3.35·25-s + 5.59·26-s − 5.71·27-s − 4.11·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.63·3-s + 0.5·4-s + 0.574·5-s − 1.15·6-s − 1.55·7-s + 0.353·8-s + 1.67·9-s + 0.405·10-s − 1.57·11-s − 0.817·12-s + 1.55·13-s − 1.10·14-s − 0.938·15-s + 0.250·16-s + 0.127·17-s + 1.18·18-s − 0.229·19-s + 0.287·20-s + 2.54·21-s − 1.11·22-s + 1.19·23-s − 0.578·24-s − 0.670·25-s + 1.09·26-s − 1.10·27-s − 0.778·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.123769570\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123769570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 0.524T + 17T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 - 0.704T + 37T^{2} \) |
| 41 | \( 1 - 8.38T + 41T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 5.63T + 59T^{2} \) |
| 61 | \( 1 - 6.12T + 61T^{2} \) |
| 67 | \( 1 - 6.03T + 67T^{2} \) |
| 71 | \( 1 + 3.00T + 71T^{2} \) |
| 73 | \( 1 + 0.00181T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 7.23T + 83T^{2} \) |
| 89 | \( 1 + 7.41T + 89T^{2} \) |
| 97 | \( 1 - 4.32T + 97T^{2} \) |
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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
−7.49921020873564656038352404137, −6.71146506110470099225222944528, −6.19859892123426573897838335074, −5.84611676237788548250742490523, −5.25337006443829533387290157182, −4.53326299457034098606564768939, −3.50690389648096126956401774102, −2.91260718462779585953111990911, −1.69450749019849274061583537129, −0.50503862182057505912831973015,
0.50503862182057505912831973015, 1.69450749019849274061583537129, 2.91260718462779585953111990911, 3.50690389648096126956401774102, 4.53326299457034098606564768939, 5.25337006443829533387290157182, 5.84611676237788548250742490523, 6.19859892123426573897838335074, 6.71146506110470099225222944528, 7.49921020873564656038352404137