| L(s) = 1 | + 8.62·3-s + 4.10·5-s − 11.2·7-s + 47.4·9-s + 71.9·11-s − 33.7·13-s + 35.4·15-s + 18.4·17-s + 19·19-s − 96.7·21-s − 86.8·23-s − 108.·25-s + 176.·27-s + 192.·29-s − 203.·31-s + 620.·33-s − 46.0·35-s + 281.·37-s − 291.·39-s − 139.·41-s − 70.7·43-s + 194.·45-s − 275.·47-s − 217.·49-s + 158.·51-s + 296.·53-s + 295.·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 0.367·5-s − 0.605·7-s + 1.75·9-s + 1.97·11-s − 0.719·13-s + 0.609·15-s + 0.262·17-s + 0.229·19-s − 1.00·21-s − 0.787·23-s − 0.865·25-s + 1.25·27-s + 1.23·29-s − 1.17·31-s + 3.27·33-s − 0.222·35-s + 1.24·37-s − 1.19·39-s − 0.532·41-s − 0.250·43-s + 0.645·45-s − 0.856·47-s − 0.633·49-s + 0.436·51-s + 0.767·53-s + 0.724·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.989773763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.989773763\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 8.62T + 27T^{2} \) |
| 5 | \( 1 - 4.10T + 125T^{2} \) |
| 7 | \( 1 + 11.2T + 343T^{2} \) |
| 11 | \( 1 - 71.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 86.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 192.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 70.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 884.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 67.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 737.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 700.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 91.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 71.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 296.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 28.5T + 9.12e5T^{2} \) |
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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
−12.72016270727663338604762981045, −11.72344544083607147635599233083, −9.873226755714305808821045373886, −9.499345128877203336813066758248, −8.528930677178514105425213913206, −7.36361699289116674150897889964, −6.26013975143462193044104847713, −4.20922659002537622058237343608, −3.16270926604489731687429460702, −1.73407631659981662052082709245,
1.73407631659981662052082709245, 3.16270926604489731687429460702, 4.20922659002537622058237343608, 6.26013975143462193044104847713, 7.36361699289116674150897889964, 8.528930677178514105425213913206, 9.499345128877203336813066758248, 9.873226755714305808821045373886, 11.72344544083607147635599233083, 12.72016270727663338604762981045
