| L(s) = 1 | + 2-s + 4-s + 0.397·5-s + 2.64·7-s + 8-s + 0.397·10-s + 4.98·11-s + 13-s + 2.64·14-s + 16-s − 2.37·17-s − 0.784·19-s + 0.397·20-s + 4.98·22-s + 6.89·23-s − 4.84·25-s + 26-s + 2.64·28-s − 2.62·29-s − 2.11·31-s + 32-s − 2.37·34-s + 1.05·35-s + 0.708·37-s − 0.784·38-s + 0.397·40-s − 5.80·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.177·5-s + 0.999·7-s + 0.353·8-s + 0.125·10-s + 1.50·11-s + 0.277·13-s + 0.707·14-s + 0.250·16-s − 0.576·17-s − 0.179·19-s + 0.0889·20-s + 1.06·22-s + 1.43·23-s − 0.968·25-s + 0.196·26-s + 0.500·28-s − 0.487·29-s − 0.380·31-s + 0.176·32-s − 0.407·34-s + 0.177·35-s + 0.116·37-s − 0.127·38-s + 0.0628·40-s − 0.906·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.569379761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.569379761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.397T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 4.98T + 11T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 + 0.784T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 - 0.708T + 37T^{2} \) |
| 41 | \( 1 + 5.80T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 4.86T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 5.77T + 83T^{2} \) |
| 89 | \( 1 - 4.51T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.978707612290009292095114449280, −8.411260745515815386492774425645, −7.31932762639747707688949380626, −6.72693029302571351163156790692, −5.85758122812382268208625172429, −5.03392552246990875808921119927, −4.22530818663473230718055805049, −3.49560133467209143574477976616, −2.15670994066978562095043291452, −1.28948537168358380415288280547,
1.28948537168358380415288280547, 2.15670994066978562095043291452, 3.49560133467209143574477976616, 4.22530818663473230718055805049, 5.03392552246990875808921119927, 5.85758122812382268208625172429, 6.72693029302571351163156790692, 7.31932762639747707688949380626, 8.411260745515815386492774425645, 8.978707612290009292095114449280
