| L(s) = 1 | + 5.03·2-s + 17.3·4-s − 0.885·5-s + 3.81·7-s + 46.9·8-s − 4.45·10-s + 52.3·11-s − 8.06·13-s + 19.2·14-s + 97.5·16-s + 17·17-s − 66.5·19-s − 15.3·20-s + 263.·22-s − 180.·23-s − 124.·25-s − 40.5·26-s + 66.1·28-s + 41.2·29-s − 34.9·31-s + 115.·32-s + 85.5·34-s − 3.38·35-s + 130.·37-s − 334.·38-s − 41.5·40-s + 17.9·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.16·4-s − 0.0792·5-s + 0.206·7-s + 2.07·8-s − 0.140·10-s + 1.43·11-s − 0.171·13-s + 0.366·14-s + 1.52·16-s + 0.242·17-s − 0.803·19-s − 0.171·20-s + 2.55·22-s − 1.63·23-s − 0.993·25-s − 0.305·26-s + 0.446·28-s + 0.264·29-s − 0.202·31-s + 0.638·32-s + 0.431·34-s − 0.0163·35-s + 0.579·37-s − 1.42·38-s − 0.164·40-s + 0.0682·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.660620220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.660620220\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 5.03T + 8T^{2} \) |
| 5 | \( 1 + 0.885T + 125T^{2} \) |
| 7 | \( 1 - 3.81T + 343T^{2} \) |
| 11 | \( 1 - 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.06T + 2.19e3T^{2} \) |
| 19 | \( 1 + 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 15.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 379.T + 9.12e5T^{2} \) |
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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
−12.48489624373236775736123338260, −11.88052770256101306632685248067, −11.02737283016101839651636562940, −9.624396572606530069461395954809, −8.002435324201369610023989484499, −6.65283483138256497126441004186, −5.87803139257014233075472377117, −4.47662001400708767625410092592, −3.66187611369757625938922168409, −1.98678942895908199373291215876,
1.98678942895908199373291215876, 3.66187611369757625938922168409, 4.47662001400708767625410092592, 5.87803139257014233075472377117, 6.65283483138256497126441004186, 8.002435324201369610023989484499, 9.624396572606530069461395954809, 11.02737283016101839651636562940, 11.88052770256101306632685248067, 12.48489624373236775736123338260
