| L(s) = 1 | + 2-s + 4-s − 2.64·5-s − 5.10·7-s + 8-s − 2.64·10-s − 2.86·11-s − 5.55·13-s − 5.10·14-s + 16-s − 1.91·19-s − 2.64·20-s − 2.86·22-s − 1.41·23-s + 2.01·25-s − 5.55·26-s − 5.10·28-s − 6.39·29-s + 2.11·31-s + 32-s + 13.5·35-s + 8.34·37-s − 1.91·38-s − 2.64·40-s − 1.01·41-s + 4.35·43-s − 2.86·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.18·5-s − 1.93·7-s + 0.353·8-s − 0.837·10-s − 0.864·11-s − 1.54·13-s − 1.36·14-s + 0.250·16-s − 0.438·19-s − 0.592·20-s − 0.611·22-s − 0.294·23-s + 0.403·25-s − 1.09·26-s − 0.965·28-s − 1.18·29-s + 0.380·31-s + 0.176·32-s + 2.28·35-s + 1.37·37-s − 0.309·38-s − 0.418·40-s − 0.159·41-s + 0.664·43-s − 0.432·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6307770991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6307770991\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 + 5.10T + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 6.39T + 29T^{2} \) |
| 31 | \( 1 - 2.11T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 4.35T + 43T^{2} \) |
| 47 | \( 1 - 4.44T + 47T^{2} \) |
| 53 | \( 1 + 5.68T + 53T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 - 0.283T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 - 0.173T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 6.54T + 83T^{2} \) |
| 89 | \( 1 + 5.43T + 89T^{2} \) |
| 97 | \( 1 - 8.57T + 97T^{2} \) |
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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
−7.76861394791293598236291270187, −7.54630764758441963906191656965, −6.75342600740130036089821128233, −6.05898476770056200003995556277, −5.27342760701581082078607708973, −4.33925437975236118215294580413, −3.77551502441467373576692131895, −2.92192630285172580805452300323, −2.40617234289318510580495011981, −0.35262164223392370374003819732,
0.35262164223392370374003819732, 2.40617234289318510580495011981, 2.92192630285172580805452300323, 3.77551502441467373576692131895, 4.33925437975236118215294580413, 5.27342760701581082078607708973, 6.05898476770056200003995556277, 6.75342600740130036089821128233, 7.54630764758441963906191656965, 7.76861394791293598236291270187
