| L(s) = 1 | − 1.56·2-s − 8.68·3-s − 5.56·4-s + 3.56·5-s + 13.5·6-s + 21.1·8-s + 48.4·9-s − 5.56·10-s + 15.2·11-s + 48.3·12-s + 13·13-s − 30.9·15-s + 11.4·16-s − 44.5·17-s − 75.6·18-s − 23.9·19-s − 19.8·20-s − 23.8·22-s + 122.·23-s − 183.·24-s − 112.·25-s − 20.3·26-s − 186.·27-s − 219.·29-s + 48.3·30-s − 27.0·31-s − 187.·32-s + ⋯ |
| L(s) = 1 | − 0.552·2-s − 1.67·3-s − 0.695·4-s + 0.318·5-s + 0.922·6-s + 0.935·8-s + 1.79·9-s − 0.175·10-s + 0.418·11-s + 1.16·12-s + 0.277·13-s − 0.532·15-s + 0.178·16-s − 0.635·17-s − 0.990·18-s − 0.289·19-s − 0.221·20-s − 0.230·22-s + 1.11·23-s − 1.56·24-s − 0.898·25-s − 0.153·26-s − 1.32·27-s − 1.40·29-s + 0.293·30-s − 0.156·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4991446300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4991446300\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 3 | \( 1 + 8.68T + 27T^{2} \) |
| 5 | \( 1 - 3.56T + 125T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 512.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−10.13877387658825147480393978269, −9.521045511277581739289449612430, −8.598576075311458985079793259617, −7.40723318258213383083132241421, −6.51070563234251516058531632578, −5.61852337632196502864439090358, −4.83480671948450839547481616176, −3.89014916760756351324370513641, −1.65998742274114912024225092113, −0.51574512209756138484841853578,
0.51574512209756138484841853578, 1.65998742274114912024225092113, 3.89014916760756351324370513641, 4.83480671948450839547481616176, 5.61852337632196502864439090358, 6.51070563234251516058531632578, 7.40723318258213383083132241421, 8.598576075311458985079793259617, 9.521045511277581739289449612430, 10.13877387658825147480393978269
