| L(s) = 1 | − 4.76·2-s + 8.84·3-s + 14.7·4-s − 7.92·5-s − 42.1·6-s − 15.4·7-s − 32.0·8-s + 51.1·9-s + 37.7·10-s + 54.2·11-s + 130.·12-s − 20.2·13-s + 73.4·14-s − 70.0·15-s + 35.0·16-s − 106.·17-s − 243.·18-s + 52.0·19-s − 116.·20-s − 136.·21-s − 258.·22-s + 30.1·23-s − 283.·24-s − 62.1·25-s + 96.6·26-s + 213.·27-s − 226.·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.70·3-s + 1.84·4-s − 0.708·5-s − 2.86·6-s − 0.831·7-s − 1.41·8-s + 1.89·9-s + 1.19·10-s + 1.48·11-s + 3.13·12-s − 0.432·13-s + 1.40·14-s − 1.20·15-s + 0.547·16-s − 1.51·17-s − 3.19·18-s + 0.628·19-s − 1.30·20-s − 1.41·21-s − 2.50·22-s + 0.273·23-s − 2.41·24-s − 0.497·25-s + 0.729·26-s + 1.52·27-s − 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - 983T \) |
| good | 2 | \( 1 + 4.76T + 8T^{2} \) |
| 3 | \( 1 - 8.84T + 27T^{2} \) |
| 5 | \( 1 + 7.92T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 11 | \( 1 - 54.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 20.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 675.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 135.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 655.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 141.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 981.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 679.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 46.6T + 9.12e5T^{2} \) |
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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
−9.003721038670217949113780905357, −8.712819629824322645307903986245, −7.74180276895967790580620630533, −7.11427883641996650202673180162, −6.48739989049732905248510932305, −4.28826130929312570449882321156, −3.44397872708895757176490698467, −2.43899683023098294175456341924, −1.43141356356836653634429543515, 0,
1.43141356356836653634429543515, 2.43899683023098294175456341924, 3.44397872708895757176490698467, 4.28826130929312570449882321156, 6.48739989049732905248510932305, 7.11427883641996650202673180162, 7.74180276895967790580620630533, 8.712819629824322645307903986245, 9.003721038670217949113780905357
