| L(s) = 1 | + 2-s − 3-s − 4-s + 0.561·5-s − 6-s − 3·8-s + 9-s + 0.561·10-s + 0.561·11-s + 12-s + 0.561·13-s − 0.561·15-s − 16-s + 18-s + 3.12·19-s − 0.561·20-s + 0.561·22-s + 23-s + 3·24-s − 4.68·25-s + 0.561·26-s − 27-s − 29-s − 0.561·30-s − 6.56·31-s + 5·32-s − 0.561·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.5·4-s + 0.251·5-s − 0.408·6-s − 1.06·8-s + 0.333·9-s + 0.177·10-s + 0.169·11-s + 0.288·12-s + 0.155·13-s − 0.144·15-s − 0.250·16-s + 0.235·18-s + 0.716·19-s − 0.125·20-s + 0.119·22-s + 0.208·23-s + 0.612·24-s − 0.936·25-s + 0.110·26-s − 0.192·27-s − 0.185·29-s − 0.102·30-s − 1.17·31-s + 0.883·32-s − 0.0977·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659746449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659746449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 0.561T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 5.43T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.434501744575075437156938208027, −8.393264683229346775536523378162, −7.49127107885491872718517548418, −6.55354468413632558976696801205, −5.67831107187774181558188780024, −5.31449948424390767156291370309, −4.23168313417245510196150790233, −3.62655488036263469667065884815, −2.35657539214597469895680509233, −0.804732735210478959261947954201,
0.804732735210478959261947954201, 2.35657539214597469895680509233, 3.62655488036263469667065884815, 4.23168313417245510196150790233, 5.31449948424390767156291370309, 5.67831107187774181558188780024, 6.55354468413632558976696801205, 7.49127107885491872718517548418, 8.393264683229346775536523378162, 9.434501744575075437156938208027
