| L(s) = 1 | + 1.43·2-s − 5.93·4-s − 5·5-s − 31.0·7-s − 20.0·8-s − 7.19·10-s + 11·11-s − 45.6·13-s − 44.6·14-s + 18.6·16-s + 40.4·17-s + 91.2·19-s + 29.6·20-s + 15.8·22-s − 32.2·23-s + 25·25-s − 65.6·26-s + 184.·28-s − 35.8·29-s + 311.·31-s + 187.·32-s + 58.1·34-s + 155.·35-s − 368.·37-s + 131.·38-s + 100.·40-s + 393.·41-s + ⋯ |
| L(s) = 1 | + 0.508·2-s − 0.741·4-s − 0.447·5-s − 1.67·7-s − 0.885·8-s − 0.227·10-s + 0.301·11-s − 0.973·13-s − 0.852·14-s + 0.290·16-s + 0.577·17-s + 1.10·19-s + 0.331·20-s + 0.153·22-s − 0.292·23-s + 0.200·25-s − 0.494·26-s + 1.24·28-s − 0.229·29-s + 1.80·31-s + 1.03·32-s + 0.293·34-s + 0.749·35-s − 1.63·37-s + 0.560·38-s + 0.396·40-s + 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.028432914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028432914\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.43T + 8T^{2} \) |
| 7 | \( 1 + 31.0T + 343T^{2} \) |
| 13 | \( 1 + 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 311.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 368.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 230.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 368.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 667.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 84.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 411.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 835.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 799.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 768.T + 9.12e5T^{2} \) |
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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
−10.21681670870553700116200423448, −9.697222859871458548097172013759, −8.940927161922406908556988339043, −7.75394748878721718647080477290, −6.73617578194218744828573251353, −5.78640098468598387169394831357, −4.74518155032672731477741009158, −3.62002124812203091282970139162, −2.93595505193767601976422913267, −0.56810332857458585681435008456,
0.56810332857458585681435008456, 2.93595505193767601976422913267, 3.62002124812203091282970139162, 4.74518155032672731477741009158, 5.78640098468598387169394831357, 6.73617578194218744828573251353, 7.75394748878721718647080477290, 8.940927161922406908556988339043, 9.697222859871458548097172013759, 10.21681670870553700116200423448
