| L(s) = 1 | + 2.49·2-s + 1.49·3-s + 4.20·4-s + 3.71·6-s + 5.49·8-s − 0.777·9-s − 3.42·11-s + 6.26·12-s + 6.26·13-s + 5.26·16-s + 5.20·17-s − 1.93·18-s + 2.22·19-s − 8.53·22-s + 2.91·23-s + 8.18·24-s − 5·25-s + 15.6·26-s − 5.63·27-s + 4.98·29-s + 0.854·31-s + 2.14·32-s − 5.10·33-s + 12.9·34-s − 3.26·36-s + 0.268·37-s + 5.53·38-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.860·3-s + 2.10·4-s + 1.51·6-s + 1.94·8-s − 0.259·9-s − 1.03·11-s + 1.80·12-s + 1.73·13-s + 1.31·16-s + 1.26·17-s − 0.456·18-s + 0.509·19-s − 1.81·22-s + 0.608·23-s + 1.67·24-s − 25-s + 3.06·26-s − 1.08·27-s + 0.925·29-s + 0.153·31-s + 0.378·32-s − 0.889·33-s + 2.22·34-s − 0.544·36-s + 0.0440·37-s + 0.898·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.768290648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.768290648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 - 6.26T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 - 0.854T + 31T^{2} \) |
| 37 | \( 1 - 0.268T + 37T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 6.12T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 8.06T + 97T^{2} \) |
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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
−9.013969299215016980233121726822, −8.108778248243266221543607618883, −7.61888424243287953806917782149, −6.46275403261896983250122787352, −5.76476478707894776188718291553, −5.16007147633462664088036963990, −4.07180317785127454839367233571, −3.23655563653654186676139364623, −2.88307662393173824627902503160, −1.58376281372782804575721838264,
1.58376281372782804575721838264, 2.88307662393173824627902503160, 3.23655563653654186676139364623, 4.07180317785127454839367233571, 5.16007147633462664088036963990, 5.76476478707894776188718291553, 6.46275403261896983250122787352, 7.61888424243287953806917782149, 8.108778248243266221543607618883, 9.013969299215016980233121726822
