| L(s) = 1 | − 1.43·2-s − 5.93·4-s + 31.0·7-s + 20.0·8-s + 11·11-s + 45.6·13-s − 44.6·14-s + 18.6·16-s − 40.4·17-s + 91.2·19-s − 15.8·22-s + 32.2·23-s − 65.6·26-s − 184.·28-s − 35.8·29-s + 311.·31-s − 187.·32-s + 58.1·34-s + 368.·37-s − 131.·38-s + 393.·41-s + 351.·43-s − 65.2·44-s − 46.3·46-s − 230.·47-s + 621.·49-s − 270.·52-s + ⋯ |
| L(s) = 1 | − 0.508·2-s − 0.741·4-s + 1.67·7-s + 0.885·8-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 − 0.153·22-s + 0.292·23-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 + 1.63·37-s − 0.560·38-s + 1.49·41-s + 1.24·43-s − 0.223·44-s − 0.148·46-s − 0.714·47-s + 1.81·49-s − 0.721·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.333617829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.333617829\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 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
−8.650642842885239865267646258897, −7.87885320910221136765631343248, −7.47217664194056023401289043326, −6.19300133631739342570347466613, −5.35671814215481847010813043386, −4.50194385270969229534979394340, −4.06776977973669095190490156770, −2.64040219117758273629868311359, −1.34262396739187894758984878955, −0.887061440225823710736722321107,
0.887061440225823710736722321107, 1.34262396739187894758984878955, 2.64040219117758273629868311359, 4.06776977973669095190490156770, 4.50194385270969229534979394340, 5.35671814215481847010813043386, 6.19300133631739342570347466613, 7.47217664194056023401289043326, 7.87885320910221136765631343248, 8.650642842885239865267646258897
