| L(s) = 1 | + 2-s + 4-s − 0.302·5-s − 1.13·7-s + 8-s − 0.302·10-s − 4.19·11-s + 2.26·13-s − 1.13·14-s + 16-s + 0.509·17-s − 0.322·19-s − 0.302·20-s − 4.19·22-s + 2.53·23-s − 4.90·25-s + 2.26·26-s − 1.13·28-s + 1.17·29-s − 3.06·31-s + 32-s + 0.509·34-s + 0.344·35-s + 7.76·37-s − 0.322·38-s − 0.302·40-s − 0.564·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.135·5-s − 0.430·7-s + 0.353·8-s − 0.0957·10-s − 1.26·11-s + 0.628·13-s − 0.304·14-s + 0.250·16-s + 0.123·17-s − 0.0739·19-s − 0.0676·20-s − 0.893·22-s + 0.528·23-s − 0.981·25-s + 0.444·26-s − 0.215·28-s + 0.217·29-s − 0.551·31-s + 0.176·32-s + 0.0873·34-s + 0.0582·35-s + 1.27·37-s − 0.0522·38-s − 0.0478·40-s − 0.0880·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.653765387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653765387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 0.302T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 - 0.509T + 17T^{2} \) |
| 19 | \( 1 + 0.322T + 19T^{2} \) |
| 23 | \( 1 - 2.53T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 0.564T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 7.37T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 - 9.20T + 79T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 - 8.10T + 97T^{2} \) |
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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
−7.898495855063145145242992027310, −7.00386111450419441625982617888, −6.40339588561172231264137707298, −5.59062207103770375017663909085, −5.17719920239395306462403337703, −4.20254949387759118501033070744, −3.57430143803087382327035291342, −2.77357094251033073786605185696, −2.04117907986972047111573735830, −0.70315539051059376390896930573,
0.70315539051059376390896930573, 2.04117907986972047111573735830, 2.77357094251033073786605185696, 3.57430143803087382327035291342, 4.20254949387759118501033070744, 5.17719920239395306462403337703, 5.59062207103770375017663909085, 6.40339588561172231264137707298, 7.00386111450419441625982617888, 7.898495855063145145242992027310
