| L(s) = 1 | + 2.69·2-s + 3-s + 5.25·4-s − 1.04·5-s + 2.69·6-s + 8.76·8-s + 9-s − 2.82·10-s + 0.180·11-s + 5.25·12-s + 4.89·13-s − 1.04·15-s + 13.1·16-s − 2.82·17-s + 2.69·18-s + 0.180·19-s − 5.51·20-s + 0.487·22-s + 23-s + 8.76·24-s − 3.89·25-s + 13.1·26-s + 27-s − 5.68·29-s − 2.82·30-s + 0.825·31-s + 17.7·32-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.62·4-s − 0.469·5-s + 1.09·6-s + 3.09·8-s + 0.333·9-s − 0.893·10-s + 0.0545·11-s + 1.51·12-s + 1.35·13-s − 0.270·15-s + 3.27·16-s − 0.685·17-s + 0.634·18-s + 0.0415·19-s − 1.23·20-s + 0.103·22-s + 0.208·23-s + 1.78·24-s − 0.779·25-s + 2.58·26-s + 0.192·27-s − 1.05·29-s − 0.515·30-s + 0.148·31-s + 3.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.952075086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.952075086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 - 0.180T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.180T + 19T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 0.825T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 0.125T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 + 9.01T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.426690022274844139254310496576, −7.52126626765937713280596345232, −7.05950739408558674657851087510, −5.99766417698264465222653080918, −5.70299703302578241965572496874, −4.30875349990281802655770615591, −4.16903477724482984047288256295, −3.27271836358225763110519275325, −2.48828328918059724846979824369, −1.44900029405395145771073045373,
1.44900029405395145771073045373, 2.48828328918059724846979824369, 3.27271836358225763110519275325, 4.16903477724482984047288256295, 4.30875349990281802655770615591, 5.70299703302578241965572496874, 5.99766417698264465222653080918, 7.05950739408558674657851087510, 7.52126626765937713280596345232, 8.426690022274844139254310496576
