| L(s) = 1 | − 2-s + 4-s − 3.37·5-s − 7-s − 8-s + 3.37·10-s + 3.37·11-s − 13-s + 14-s + 16-s + 1.37·17-s + 7.37·19-s − 3.37·20-s − 3.37·22-s − 9.37·23-s + 6.37·25-s + 26-s − 28-s − 0.627·29-s + 6.74·31-s − 32-s − 1.37·34-s + 3.37·35-s + 1.37·37-s − 7.37·38-s + 3.37·40-s + 2.74·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.50·5-s − 0.377·7-s − 0.353·8-s + 1.06·10-s + 1.01·11-s − 0.277·13-s + 0.267·14-s + 0.250·16-s + 0.332·17-s + 1.69·19-s − 0.754·20-s − 0.718·22-s − 1.95·23-s + 1.27·25-s + 0.196·26-s − 0.188·28-s − 0.116·29-s + 1.21·31-s − 0.176·32-s − 0.235·34-s + 0.570·35-s + 0.225·37-s − 1.19·38-s + 0.533·40-s + 0.428·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 + 9.37T + 23T^{2} \) |
| 29 | \( 1 + 0.627T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 + 2.62T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.987704821613918900132439242807, −7.971201297115256601396982665464, −7.70656538507678598770762637755, −6.77507242625469800641900267219, −5.96831913046664675574752327681, −4.64992976034127445628747307556, −3.74282880291515877863947741562, −3.00834552043087154785451106595, −1.35575343499617005022179923724, 0,
1.35575343499617005022179923724, 3.00834552043087154785451106595, 3.74282880291515877863947741562, 4.64992976034127445628747307556, 5.96831913046664675574752327681, 6.77507242625469800641900267219, 7.70656538507678598770762637755, 7.971201297115256601396982665464, 8.987704821613918900132439242807
