| L(s) = 1 | + 1.74·2-s + 1.04·4-s + 3.84·5-s + 0.0845·7-s − 1.66·8-s + 6.71·10-s + 0.570·11-s + 1.00·13-s + 0.147·14-s − 4.99·16-s − 6.52·17-s + 0.290·19-s + 4.03·20-s + 0.995·22-s + 1.26·23-s + 9.78·25-s + 1.75·26-s + 0.0886·28-s + 8.61·29-s + 4.67·31-s − 5.40·32-s − 11.3·34-s + 0.324·35-s + 8.22·37-s + 0.506·38-s − 6.38·40-s − 1.13·41-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.524·4-s + 1.71·5-s + 0.0319·7-s − 0.587·8-s + 2.12·10-s + 0.171·11-s + 0.279·13-s + 0.0394·14-s − 1.24·16-s − 1.58·17-s + 0.0665·19-s + 0.901·20-s + 0.212·22-s + 0.264·23-s + 1.95·25-s + 0.345·26-s + 0.0167·28-s + 1.60·29-s + 0.839·31-s − 0.955·32-s − 1.95·34-s + 0.0549·35-s + 1.35·37-s + 0.0822·38-s − 1.00·40-s − 0.176·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.168495614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.168495614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 1.74T + 2T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 - 0.0845T + 7T^{2} \) |
| 11 | \( 1 - 0.570T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.290T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 - 8.22T + 37T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 - 0.577T + 59T^{2} \) |
| 61 | \( 1 + 0.979T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 0.781T + 97T^{2} \) |
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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
−8.144920728609533060736843026625, −6.85677822069153306996556764672, −6.31164727908470374212806883776, −6.05007929807818358850268217345, −5.03785806974886250046118100129, −4.68824387003917195985208422261, −3.78235970136419220769968817051, −2.59956573882943896425229794385, −2.36113663108498851581384763460, −1.02651563571193258793755216938,
1.02651563571193258793755216938, 2.36113663108498851581384763460, 2.59956573882943896425229794385, 3.78235970136419220769968817051, 4.68824387003917195985208422261, 5.03785806974886250046118100129, 6.05007929807818358850268217345, 6.31164727908470374212806883776, 6.85677822069153306996556764672, 8.144920728609533060736843026625
