| L(s) = 1 | + 2-s − 1.89·3-s + 4-s + 0.311·5-s − 1.89·6-s + 2.23·7-s + 8-s + 0.607·9-s + 0.311·10-s + 4.25·11-s − 1.89·12-s + 2.41·13-s + 2.23·14-s − 0.592·15-s + 16-s + 6.75·17-s + 0.607·18-s − 6.91·19-s + 0.311·20-s − 4.24·21-s + 4.25·22-s + 5.68·23-s − 1.89·24-s − 4.90·25-s + 2.41·26-s + 4.54·27-s + 2.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.09·3-s + 0.5·4-s + 0.139·5-s − 0.775·6-s + 0.844·7-s + 0.353·8-s + 0.202·9-s + 0.0985·10-s + 1.28·11-s − 0.548·12-s + 0.668·13-s + 0.597·14-s − 0.152·15-s + 0.250·16-s + 1.63·17-s + 0.143·18-s − 1.58·19-s + 0.0696·20-s − 0.926·21-s + 0.906·22-s + 1.18·23-s − 0.387·24-s − 0.980·25-s + 0.472·26-s + 0.874·27-s + 0.422·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.711160669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.711160669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 - 0.311T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 4.25T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 - 6.13T + 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 - 3.90T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 - 2.94T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.07T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 8.30T + 97T^{2} \) |
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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
−8.413675987129555168792648383328, −7.53122412479638105353003659690, −6.63468457608604173126369711853, −6.11938895631754081224086691848, −5.51344486674527073157007696371, −4.76046387341028939813392622668, −4.05844351836252169246301488031, −3.16259106782990237349280871845, −1.79839438897109034819143274398, −0.971628603658104391461257049079,
0.971628603658104391461257049079, 1.79839438897109034819143274398, 3.16259106782990237349280871845, 4.05844351836252169246301488031, 4.76046387341028939813392622668, 5.51344486674527073157007696371, 6.11938895631754081224086691848, 6.63468457608604173126369711853, 7.53122412479638105353003659690, 8.413675987129555168792648383328
