| L(s) = 1 | + 1.43·2-s − 0.561·3-s − 5.93·4-s − 0.807·6-s + 31.0·7-s − 20.0·8-s − 26.6·9-s − 11·11-s + 3.33·12-s + 45.6·13-s + 44.6·14-s + 18.6·16-s + 40.4·17-s − 38.3·18-s + 91.2·19-s − 17.4·21-s − 15.8·22-s − 32.2·23-s + 11.2·24-s + 65.6·26-s + 30.1·27-s − 184.·28-s + 35.8·29-s + 311.·31-s + 187.·32-s + 6.17·33-s + 58.1·34-s + ⋯ |
| L(s) = 1 | + 0.508·2-s − 0.108·3-s − 0.741·4-s − 0.0549·6-s + 1.67·7-s − 0.885·8-s − 0.988·9-s − 0.301·11-s + 0.0801·12-s + 0.973·13-s + 0.852·14-s + 0.290·16-s + 0.577·17-s − 0.502·18-s + 1.10·19-s − 0.181·21-s − 0.153·22-s − 0.292·23-s + 0.0957·24-s + 0.494·26-s + 0.214·27-s − 1.24·28-s + 0.229·29-s + 1.80·31-s + 1.03·32-s + 0.0325·33-s + 0.293·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.110753196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.110753196\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 1.43T + 8T^{2} \) |
| 3 | \( 1 + 0.561T + 27T^{2} \) |
| 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
−11.59803719714915946088371471098, −10.73996577671307568473371378291, −9.449723296235921756473456115223, −8.346354681001119401122685413277, −7.928979957336453317481974079833, −6.03997994047472442566270334482, −5.25152315439754542348016722924, −4.35860796720789923867682379772, −2.96533231805879633072178280893, −1.04069971571058297044101241945,
1.04069971571058297044101241945, 2.96533231805879633072178280893, 4.35860796720789923867682379772, 5.25152315439754542348016722924, 6.03997994047472442566270334482, 7.928979957336453317481974079833, 8.346354681001119401122685413277, 9.449723296235921756473456115223, 10.73996577671307568473371378291, 11.59803719714915946088371471098
