| L(s) = 1 | + 2-s − 3-s + 4-s + 0.561·5-s − 6-s + 8-s + 9-s + 0.561·10-s − 2.56·11-s − 12-s + 4.56·13-s − 0.561·15-s + 16-s − 7.12·17-s + 18-s − 4·19-s + 0.561·20-s − 2.56·22-s − 23-s − 24-s − 4.68·25-s + 4.56·26-s − 27-s + 29-s − 0.561·30-s + 1.43·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.251·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.177·10-s − 0.772·11-s − 0.288·12-s + 1.26·13-s − 0.144·15-s + 0.250·16-s − 1.72·17-s + 0.235·18-s − 0.917·19-s + 0.125·20-s − 0.546·22-s − 0.208·23-s − 0.204·24-s − 0.936·25-s + 0.894·26-s − 0.192·27-s + 0.185·29-s − 0.102·30-s + 0.258·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 3.68T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.135354255504604244100306391314, −7.03372647828807455517977395877, −6.42674187093949108722471056831, −5.91148620702210736714497553074, −5.06933835668419336715642125249, −4.35371340288753472774717093933, −3.61254654267315540954608392109, −2.45079279650661634717564003553, −1.63248556203682093007332564361, 0,
1.63248556203682093007332564361, 2.45079279650661634717564003553, 3.61254654267315540954608392109, 4.35371340288753472774717093933, 5.06933835668419336715642125249, 5.91148620702210736714497553074, 6.42674187093949108722471056831, 7.03372647828807455517977395877, 8.135354255504604244100306391314
