| L(s) = 1 | − 2.34·2-s − 0.698·3-s + 3.47·4-s + 4.27·5-s + 1.63·6-s − 0.686·7-s − 3.46·8-s − 2.51·9-s − 10.0·10-s − 1.00·11-s − 2.42·12-s − 3.40·13-s + 1.60·14-s − 2.98·15-s + 1.14·16-s − 6.21·17-s + 5.88·18-s − 3.26·19-s + 14.8·20-s + 0.479·21-s + 2.35·22-s + 0.141·23-s + 2.41·24-s + 13.2·25-s + 7.97·26-s + 3.84·27-s − 2.38·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 0.403·3-s + 1.73·4-s + 1.91·5-s + 0.667·6-s − 0.259·7-s − 1.22·8-s − 0.837·9-s − 3.16·10-s − 0.303·11-s − 0.701·12-s − 0.944·13-s + 0.429·14-s − 0.770·15-s + 0.286·16-s − 1.50·17-s + 1.38·18-s − 0.748·19-s + 3.32·20-s + 0.104·21-s + 0.502·22-s + 0.0295·23-s + 0.493·24-s + 2.65·25-s + 1.56·26-s + 0.740·27-s − 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5733092726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5733092726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 + 0.698T + 3T^{2} \) |
| 5 | \( 1 - 4.27T + 5T^{2} \) |
| 7 | \( 1 + 0.686T + 7T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 0.141T + 23T^{2} \) |
| 29 | \( 1 + 0.745T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 + 0.422T + 37T^{2} \) |
| 41 | \( 1 - 6.86T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 0.269T + 67T^{2} \) |
| 73 | \( 1 - 7.27T + 73T^{2} \) |
| 79 | \( 1 - 3.19T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.115612195813996586420391957265, −6.97910783260140148735556991559, −6.65040665114882561616361251050, −5.99907104599312775493510983664, −5.32547481710158322737293073709, −4.54391123725994184196185917038, −2.83996917558674293556084074638, −2.32166591748453196383720801236, −1.73797585495444913301623901758, −0.47000235447903319553190248109,
0.47000235447903319553190248109, 1.73797585495444913301623901758, 2.32166591748453196383720801236, 2.83996917558674293556084074638, 4.54391123725994184196185917038, 5.32547481710158322737293073709, 5.99907104599312775493510983664, 6.65040665114882561616361251050, 6.97910783260140148735556991559, 8.115612195813996586420391957265
