| L(s) = 1 | − 1.98·2-s + 1.92·4-s + 0.252·5-s − 1.21·7-s + 0.138·8-s − 0.501·10-s − 2.73·11-s − 6.01·13-s + 2.40·14-s − 4.13·16-s − 0.539·17-s + 3.87·19-s + 0.488·20-s + 5.43·22-s + 23-s − 4.93·25-s + 11.9·26-s − 2.34·28-s + 29-s − 10.4·31-s + 7.91·32-s + 1.06·34-s − 0.307·35-s − 4.48·37-s − 7.68·38-s + 0.0351·40-s + 7.22·41-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 0.964·4-s + 0.113·5-s − 0.459·7-s + 0.0491·8-s − 0.158·10-s − 0.826·11-s − 1.66·13-s + 0.643·14-s − 1.03·16-s − 0.130·17-s + 0.889·19-s + 0.109·20-s + 1.15·22-s + 0.208·23-s − 0.987·25-s + 2.33·26-s − 0.443·28-s + 0.185·29-s − 1.87·31-s + 1.40·32-s + 0.183·34-s − 0.0519·35-s − 0.737·37-s − 1.24·38-s + 0.00555·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3237907374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3237907374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 5 | \( 1 - 0.252T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 + 0.539T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 4.48T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 0.281T + 73T^{2} \) |
| 79 | \( 1 + 9.98T + 79T^{2} \) |
| 83 | \( 1 - 0.778T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.947997445079431955117452190915, −7.51178311648492701944995093551, −7.12931537827352040816438809341, −6.10139189553256184599253371714, −5.24078444752808252374323110455, −4.59180960720731687052516395233, −3.37989223930001753817980394060, −2.47540356786998306665057585483, −1.72819578611237098670166914272, −0.35758276752957612552772851501,
0.35758276752957612552772851501, 1.72819578611237098670166914272, 2.47540356786998306665057585483, 3.37989223930001753817980394060, 4.59180960720731687052516395233, 5.24078444752808252374323110455, 6.10139189553256184599253371714, 7.12931537827352040816438809341, 7.51178311648492701944995093551, 7.947997445079431955117452190915
