| L(s) = 1 | − 2-s + 1.53·5-s + 8-s + 9-s − 1.53·10-s − 1.87·11-s + 1.53·13-s − 16-s − 18-s + 1.87·22-s + 1.53·23-s + 1.34·25-s − 1.53·26-s + 0.347·31-s − 1.87·37-s + 1.53·40-s − 41-s + 1.53·45-s − 1.53·46-s + 49-s − 1.34·50-s + 0.347·53-s − 2.87·55-s − 1.87·59-s − 1.87·61-s − 0.347·62-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s + 1.53·5-s + 8-s + 9-s − 1.53·10-s − 1.87·11-s + 1.53·13-s − 16-s − 18-s + 1.87·22-s + 1.53·23-s + 1.34·25-s − 1.53·26-s + 0.347·31-s − 1.87·37-s + 1.53·40-s − 41-s + 1.53·45-s − 1.53·46-s + 49-s − 1.34·50-s + 0.347·53-s − 2.87·55-s − 1.87·59-s − 1.87·61-s − 0.347·62-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8994269410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8994269410\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1999 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.87T + T^{2} \) |
| 13 | \( 1 - 1.53T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.53T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.347T + T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 + 1.87T + T^{2} \) |
| 61 | \( 1 + 1.87T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.347T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.237030948978955279447246654501, −8.818440411410533252646049555657, −7.937326921763223739114569553317, −7.15423295814401601600265974331, −6.29433398721245465103689290350, −5.30372617148374457722512250537, −4.76516615331037771528884451315, −3.30465269527798867551458566595, −2.05696817248293942945760028354, −1.22651824257566684339351614640,
1.22651824257566684339351614640, 2.05696817248293942945760028354, 3.30465269527798867551458566595, 4.76516615331037771528884451315, 5.30372617148374457722512250537, 6.29433398721245465103689290350, 7.15423295814401601600265974331, 7.937326921763223739114569553317, 8.818440411410533252646049555657, 9.237030948978955279447246654501
