| L(s) = 1 | + 2-s + 4-s + 3.73·5-s − 2.62·7-s + 8-s + 3.73·10-s − 2.14·11-s − 0.987·13-s − 2.62·14-s + 16-s − 5.24·17-s − 4.24·19-s + 3.73·20-s − 2.14·22-s − 8.14·23-s + 8.94·25-s − 0.987·26-s − 2.62·28-s + 8.43·29-s − 5.21·31-s + 32-s − 5.24·34-s − 9.79·35-s + 0.533·37-s − 4.24·38-s + 3.73·40-s + 3.88·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.66·5-s − 0.991·7-s + 0.353·8-s + 1.18·10-s − 0.646·11-s − 0.273·13-s − 0.701·14-s + 0.250·16-s − 1.27·17-s − 0.973·19-s + 0.834·20-s − 0.456·22-s − 1.69·23-s + 1.78·25-s − 0.193·26-s − 0.495·28-s + 1.56·29-s − 0.937·31-s + 0.176·32-s − 0.899·34-s − 1.65·35-s + 0.0877·37-s − 0.688·38-s + 0.590·40-s + 0.606·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 + 0.987T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 8.14T + 23T^{2} \) |
| 29 | \( 1 - 8.43T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 - 0.533T + 37T^{2} \) |
| 41 | \( 1 - 3.88T + 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 0.811T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 + 8.88T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 6.07T + 89T^{2} \) |
| 97 | \( 1 + 1.67T + 97T^{2} \) |
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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
−7.18927503637289057762104067342, −6.42990134855024752043360267975, −6.16255231938810445175871424445, −5.54889664040477368345243201554, −4.68401001085810609297009758974, −4.03429363976114872113848756646, −2.83084891057904710328697331989, −2.42738892011652892843306076406, −1.66057319853181183871005910212, 0,
1.66057319853181183871005910212, 2.42738892011652892843306076406, 2.83084891057904710328697331989, 4.03429363976114872113848756646, 4.68401001085810609297009758974, 5.54889664040477368345243201554, 6.16255231938810445175871424445, 6.42990134855024752043360267975, 7.18927503637289057762104067342
