| L(s) = 1 | − 0.228·2-s − 1.94·4-s − 0.872·5-s − 1.27·7-s + 0.902·8-s + 0.199·10-s − 11-s + 5.51·13-s + 0.291·14-s + 3.68·16-s + 7.05·17-s − 4.53·19-s + 1.69·20-s + 0.228·22-s − 4.97·23-s − 4.23·25-s − 1.26·26-s + 2.48·28-s + 4.43·29-s + 6.49·31-s − 2.64·32-s − 1.61·34-s + 1.11·35-s + 4.65·37-s + 1.03·38-s − 0.787·40-s − 2.47·41-s + ⋯ |
| L(s) = 1 | − 0.161·2-s − 0.973·4-s − 0.390·5-s − 0.481·7-s + 0.319·8-s + 0.0630·10-s − 0.301·11-s + 1.53·13-s + 0.0779·14-s + 0.922·16-s + 1.71·17-s − 1.04·19-s + 0.379·20-s + 0.0487·22-s − 1.03·23-s − 0.847·25-s − 0.247·26-s + 0.469·28-s + 0.822·29-s + 1.16·31-s − 0.468·32-s − 0.276·34-s + 0.187·35-s + 0.765·37-s + 0.168·38-s − 0.124·40-s − 0.387·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076835889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076835889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.228T + 2T^{2} \) |
| 5 | \( 1 + 0.872T + 5T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 - 7.05T + 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 + 0.452T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 - 0.139T + 59T^{2} \) |
| 67 | \( 1 - 3.75T + 67T^{2} \) |
| 71 | \( 1 - 9.37T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 + 8.77T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.260674386593090324603522141540, −7.69449256020792178744518345222, −6.45438424005452146411810436419, −6.03561942339330448281301776013, −5.18894910249783541069745779456, −4.30197935491182093816712398169, −3.69974071774155039610657928009, −3.05779940188547895723384704876, −1.60647413490426023020203731518, −0.58818791151475540764067083505,
0.58818791151475540764067083505, 1.60647413490426023020203731518, 3.05779940188547895723384704876, 3.69974071774155039610657928009, 4.30197935491182093816712398169, 5.18894910249783541069745779456, 6.03561942339330448281301776013, 6.45438424005452146411810436419, 7.69449256020792178744518345222, 8.260674386593090324603522141540
