| L(s) = 1 | + 2-s + 4-s − 4.35·7-s + 8-s + 4.35·11-s − 4.35·14-s + 16-s + 4·17-s + 6·19-s + 4.35·22-s − 2·23-s − 4.35·28-s + 7·31-s + 32-s + 4·34-s + 8.71·37-s + 6·38-s − 8.71·41-s + 8.71·43-s + 4.35·44-s − 2·46-s + 2·47-s + 12.0·49-s − 3·53-s − 4.35·56-s + 8.71·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.64·7-s + 0.353·8-s + 1.31·11-s − 1.16·14-s + 0.250·16-s + 0.970·17-s + 1.37·19-s + 0.929·22-s − 0.417·23-s − 0.823·28-s + 1.25·31-s + 0.176·32-s + 0.685·34-s + 1.43·37-s + 0.973·38-s − 1.36·41-s + 1.32·43-s + 0.657·44-s − 0.294·46-s + 0.291·47-s + 1.71·49-s − 0.412·53-s − 0.582·56-s + 1.13·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.465816266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465816266\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 8.71T + 37T^{2} \) |
| 41 | \( 1 + 8.71T + 41T^{2} \) |
| 43 | \( 1 - 8.71T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4.35T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 + 8.71T + 89T^{2} \) |
| 97 | \( 1 - 4.35T + 97T^{2} \) |
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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
−9.745710883923792087820376688981, −8.999396774747180827227301432003, −7.76977121011768601913358247835, −6.92491019420454483770102890154, −6.23668807634405041476143501819, −5.60978920086988969726692862806, −4.31693309753682224821546583327, −3.48883838002187822386271183003, −2.80562676525849665288697615059, −1.09564415871161753804012423175,
1.09564415871161753804012423175, 2.80562676525849665288697615059, 3.48883838002187822386271183003, 4.31693309753682224821546583327, 5.60978920086988969726692862806, 6.23668807634405041476143501819, 6.92491019420454483770102890154, 7.76977121011768601913358247835, 8.999396774747180827227301432003, 9.745710883923792087820376688981
