| L(s) = 1 | − 2.21·2-s − 3-s + 2.91·4-s + 5-s + 2.21·6-s + 3.34·7-s − 2.03·8-s + 9-s − 2.21·10-s − 2.54·11-s − 2.91·12-s − 3.51·13-s − 7.41·14-s − 15-s − 1.31·16-s + 0.694·17-s − 2.21·18-s − 1.13·19-s + 2.91·20-s − 3.34·21-s + 5.64·22-s − 8.45·23-s + 2.03·24-s + 25-s + 7.80·26-s − 27-s + 9.75·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.45·4-s + 0.447·5-s + 0.905·6-s + 1.26·7-s − 0.720·8-s + 0.333·9-s − 0.701·10-s − 0.766·11-s − 0.842·12-s − 0.976·13-s − 1.98·14-s − 0.258·15-s − 0.329·16-s + 0.168·17-s − 0.522·18-s − 0.261·19-s + 0.652·20-s − 0.729·21-s + 1.20·22-s − 1.76·23-s + 0.415·24-s + 0.200·25-s + 1.53·26-s − 0.192·27-s + 1.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 - 0.694T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 + 8.45T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 1.18T + 41T^{2} \) |
| 43 | \( 1 - 6.62T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.85T + 67T^{2} \) |
| 71 | \( 1 + 0.568T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 + 8.09T + 89T^{2} \) |
| 97 | \( 1 - 1.72T + 97T^{2} \) |
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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
−8.004853999884058899510185670254, −7.35738833631864570528391523305, −6.43717380006918268956372571895, −5.76148520352896251204209089835, −4.84494352485782644414425089433, −4.35163485308264234954729294503, −2.63659306687604127969899872652, −2.02007545298389649044952521653, −1.13536841667909517130118881474, 0,
1.13536841667909517130118881474, 2.02007545298389649044952521653, 2.63659306687604127969899872652, 4.35163485308264234954729294503, 4.84494352485782644414425089433, 5.76148520352896251204209089835, 6.43717380006918268956372571895, 7.35738833631864570528391523305, 8.004853999884058899510185670254
