| L(s) = 1 | + 2-s + 3-s + 4-s + 3.05·5-s + 6-s + 7-s + 8-s + 9-s + 3.05·10-s − 2.34·11-s + 12-s + 5.11·13-s + 14-s + 3.05·15-s + 16-s + 0.266·17-s + 18-s − 0.779·19-s + 3.05·20-s + 21-s − 2.34·22-s − 0.510·23-s + 24-s + 4.32·25-s + 5.11·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.36·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.965·10-s − 0.706·11-s + 0.288·12-s + 1.41·13-s + 0.267·14-s + 0.788·15-s + 0.250·16-s + 0.0646·17-s + 0.235·18-s − 0.178·19-s + 0.682·20-s + 0.218·21-s − 0.499·22-s − 0.106·23-s + 0.204·24-s + 0.864·25-s + 1.00·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.140956916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.140956916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 - 3.05T + 5T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 - 0.266T + 17T^{2} \) |
| 19 | \( 1 + 0.779T + 19T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 1.47T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 0.946T + 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.88318575738183255268525710013, −6.99411380088793969267253066348, −6.33606787897966555824058664037, −5.48856898104189141221071096960, −5.38460131676336112398863265549, −4.12857271609670152805853784507, −3.60939338509529544800019720447, −2.52715084329744669883783639406, −2.05115480716486307915365318341, −1.16058500705562672531335690178,
1.16058500705562672531335690178, 2.05115480716486307915365318341, 2.52715084329744669883783639406, 3.60939338509529544800019720447, 4.12857271609670152805853784507, 5.38460131676336112398863265549, 5.48856898104189141221071096960, 6.33606787897966555824058664037, 6.99411380088793969267253066348, 7.88318575738183255268525710013
