L(s) = 1 | + 2-s + 1.77·3-s + 4-s + 0.642·5-s + 1.77·6-s + 4.80·7-s + 8-s + 0.150·9-s + 0.642·10-s + 3.51·11-s + 1.77·12-s − 0.225·13-s + 4.80·14-s + 1.14·15-s + 16-s + 0.444·17-s + 0.150·18-s − 19-s + 0.642·20-s + 8.52·21-s + 3.51·22-s − 2.47·23-s + 1.77·24-s − 4.58·25-s − 0.225·26-s − 5.05·27-s + 4.80·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.02·3-s + 0.5·4-s + 0.287·5-s + 0.724·6-s + 1.81·7-s + 0.353·8-s + 0.0501·9-s + 0.203·10-s + 1.05·11-s + 0.512·12-s − 0.0624·13-s + 1.28·14-s + 0.294·15-s + 0.250·16-s + 0.107·17-s + 0.0354·18-s − 0.229·19-s + 0.143·20-s + 1.86·21-s + 0.748·22-s − 0.515·23-s + 0.362·24-s − 0.917·25-s − 0.0441·26-s − 0.973·27-s + 0.908·28-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
\(L(1)\) |
\(\approx\) |
\(6.826708280\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.826708280\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.77T + 3T^{2} \) |
| 5 | \( 1 - 0.642T + 5T^{2} \) |
| 7 | \( 1 - 4.80T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 + 0.225T + 13T^{2} \) |
| 17 | \( 1 - 0.444T + 17T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 0.426T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 + 6.26T + 41T^{2} \) |
| 43 | \( 1 + 6.25T + 43T^{2} \) |
| 47 | \( 1 - 2.41T + 47T^{2} \) |
| 53 | \( 1 - 0.377T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 9.01T + 71T^{2} \) |
| 73 | \( 1 + 8.40T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 5.87T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 + 3.02T + 97T^{2} \) |
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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
−7.907883930007513882105734665407, −7.25692933292939019737911762749, −6.36603455899288883982085940766, −5.67390363635678623964894993760, −4.88783346533649690493500340926, −4.22868517980449106912700972670, −3.64230013976414027432634195884, −2.59201802361864995181463816665, −1.97692176485767944951250947861, −1.24596620058221718437775254843,
1.24596620058221718437775254843, 1.97692176485767944951250947861, 2.59201802361864995181463816665, 3.64230013976414027432634195884, 4.22868517980449106912700972670, 4.88783346533649690493500340926, 5.67390363635678623964894993760, 6.36603455899288883982085940766, 7.25692933292939019737911762749, 7.907883930007513882105734665407