| L(s) = 1 | − 3·3-s + 4.43·7-s + 9·9-s + 3.43·11-s + 78.7·13-s + 53.1·17-s − 20.4·19-s − 13.3·21-s − 118.·23-s − 27·27-s + 168.·29-s + 61.0·31-s − 10.3·33-s − 246.·37-s − 236.·39-s + 422.·41-s − 362.·43-s + 170.·47-s − 323.·49-s − 159.·51-s − 546.·53-s + 61.3·57-s + 216.·59-s + 130.·61-s + 39.9·63-s + 614.·67-s + 354.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.239·7-s + 0.333·9-s + 0.0941·11-s + 1.67·13-s + 0.758·17-s − 0.246·19-s − 0.138·21-s − 1.07·23-s − 0.192·27-s + 1.07·29-s + 0.353·31-s − 0.0543·33-s − 1.09·37-s − 0.969·39-s + 1.60·41-s − 1.28·43-s + 0.529·47-s − 0.942·49-s − 0.438·51-s − 1.41·53-s + 0.142·57-s + 0.478·59-s + 0.274·61-s + 0.0798·63-s + 1.12·67-s + 0.619·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.940679970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.940679970\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.43T + 343T^{2} \) |
| 11 | \( 1 - 3.43T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 246.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 362.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 614.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 324.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 758.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 521T + 9.12e5T^{2} \) |
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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
−9.436668028118038519227309944081, −8.417040653075729077707211503908, −7.88771010992851787689019347176, −6.63703279491758345981701194380, −6.10321999594169619831863839845, −5.19955176561685212083681972357, −4.17724365826440078712774198803, −3.30053302366629128402217145378, −1.78380369826522596803033591672, −0.77105254202752678220666731640,
0.77105254202752678220666731640, 1.78380369826522596803033591672, 3.30053302366629128402217145378, 4.17724365826440078712774198803, 5.19955176561685212083681972357, 6.10321999594169619831863839845, 6.63703279491758345981701194380, 7.88771010992851787689019347176, 8.417040653075729077707211503908, 9.436668028118038519227309944081
