| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 3.77·11-s − 5.77·13-s − 14-s + 16-s + 3.77·17-s + 2·19-s + 3.77·22-s − 3.77·23-s − 5.77·26-s − 28-s − 3·29-s − 8.54·31-s + 32-s + 3.77·34-s − 8.77·37-s + 2·38-s − 6.77·41-s + 1.77·43-s + 3.77·44-s − 3.77·46-s + 4.54·47-s + 49-s − 5.77·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s + 1.13·11-s − 1.60·13-s − 0.267·14-s + 0.250·16-s + 0.914·17-s + 0.458·19-s + 0.804·22-s − 0.786·23-s − 1.13·26-s − 0.188·28-s − 0.557·29-s − 1.53·31-s + 0.176·32-s + 0.646·34-s − 1.44·37-s + 0.324·38-s − 1.05·41-s + 0.270·43-s + 0.568·44-s − 0.556·46-s + 0.662·47-s + 0.142·49-s − 0.800·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + 5.77T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.77T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 8.54T + 31T^{2} \) |
| 37 | \( 1 + 8.77T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.77T + 59T^{2} \) |
| 61 | \( 1 - 2.77T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.34397163531188470822349552269, −6.68032314305826179691084203169, −5.86139750091824762517072626030, −5.32308848756080262249245670988, −4.59532894084057359131903237471, −3.72126367223035773439927556616, −3.28140106864940852535828439820, −2.24037357195407306190476100764, −1.46229606076527616179809975547, 0,
1.46229606076527616179809975547, 2.24037357195407306190476100764, 3.28140106864940852535828439820, 3.72126367223035773439927556616, 4.59532894084057359131903237471, 5.32308848756080262249245670988, 5.86139750091824762517072626030, 6.68032314305826179691084203169, 7.34397163531188470822349552269
