| L(s) = 1 | + 2-s + 3-s + 4-s − 4.21·5-s + 6-s + 1.46·7-s + 8-s + 9-s − 4.21·10-s + 2.34·11-s + 12-s − 5.14·13-s + 1.46·14-s − 4.21·15-s + 16-s + 17-s + 18-s + 3.77·19-s − 4.21·20-s + 1.46·21-s + 2.34·22-s + 0.976·23-s + 24-s + 12.7·25-s − 5.14·26-s + 27-s + 1.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.88·5-s + 0.408·6-s + 0.553·7-s + 0.353·8-s + 0.333·9-s − 1.33·10-s + 0.708·11-s + 0.288·12-s − 1.42·13-s + 0.391·14-s − 1.08·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.865·19-s − 0.942·20-s + 0.319·21-s + 0.500·22-s + 0.203·23-s + 0.204·24-s + 2.55·25-s − 1.00·26-s + 0.192·27-s + 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.851110477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.851110477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + 4.21T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 - 0.976T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 + 8.54T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−7.78670471760993399068236715534, −7.34733344823641156128461575747, −7.10377797369805882260487661858, −5.80306861534690108906345938367, −4.85994490003051082779623392872, −4.40836702216966016116342308906, −3.65640608044449546167162557215, −3.10936637992372132500457676020, −2.07761591583593664141337288225, −0.77487013741586191296658911441,
0.77487013741586191296658911441, 2.07761591583593664141337288225, 3.10936637992372132500457676020, 3.65640608044449546167162557215, 4.40836702216966016116342308906, 4.85994490003051082779623392872, 5.80306861534690108906345938367, 7.10377797369805882260487661858, 7.34733344823641156128461575747, 7.78670471760993399068236715534
