| L(s) = 1 | − 1.61·2-s − 2.23·3-s + 0.618·4-s + 3.23·5-s + 3.61·6-s + 2.23·8-s + 2.00·9-s − 5.23·10-s − 0.763·11-s − 1.38·12-s − 3·13-s − 7.23·15-s − 4.85·16-s − 5.23·17-s − 3.23·18-s + 2·19-s + 2.00·20-s + 1.23·22-s + 23-s − 5.00·24-s + 5.47·25-s + 4.85·26-s + 2.23·27-s − 3·29-s + 11.7·30-s + 6.70·31-s + 3.38·32-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 1.29·3-s + 0.309·4-s + 1.44·5-s + 1.47·6-s + 0.790·8-s + 0.666·9-s − 1.65·10-s − 0.230·11-s − 0.398·12-s − 0.832·13-s − 1.86·15-s − 1.21·16-s − 1.26·17-s − 0.762·18-s + 0.458·19-s + 0.447·20-s + 0.263·22-s + 0.208·23-s − 1.02·24-s + 1.09·25-s + 0.951·26-s + 0.430·27-s − 0.557·29-s + 2.13·30-s + 1.20·31-s + 0.597·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−9.540160292717530504520209483437, −8.896073805332539415172836995671, −7.80318510025201343803382662436, −6.78748203020737630363524287330, −6.19006065606249903721036794227, −5.18432731114980008118481604090, −4.62622348481158235585086916903, −2.52669220302781237031827709058, −1.41011247276700253835474568164, 0,
1.41011247276700253835474568164, 2.52669220302781237031827709058, 4.62622348481158235585086916903, 5.18432731114980008118481604090, 6.19006065606249903721036794227, 6.78748203020737630363524287330, 7.80318510025201343803382662436, 8.896073805332539415172836995671, 9.540160292717530504520209483437
