| L(s) = 1 | + 2-s − 3-s − 4-s − 3.56·5-s − 6-s − 3·8-s + 9-s − 3.56·10-s − 3.56·11-s + 12-s − 3.56·13-s + 3.56·15-s − 16-s + 18-s − 5.12·19-s + 3.56·20-s − 3.56·22-s + 23-s + 3·24-s + 7.68·25-s − 3.56·26-s − 27-s − 29-s + 3.56·30-s − 2.43·31-s + 5·32-s + 3.56·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.5·4-s − 1.59·5-s − 0.408·6-s − 1.06·8-s + 0.333·9-s − 1.12·10-s − 1.07·11-s + 0.288·12-s − 0.987·13-s + 0.919·15-s − 0.250·16-s + 0.235·18-s − 1.17·19-s + 0.796·20-s − 0.759·22-s + 0.208·23-s + 0.612·24-s + 1.53·25-s − 0.698·26-s − 0.192·27-s − 0.185·29-s + 0.650·30-s − 0.437·31-s + 0.883·32-s + 0.619·33-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
| \(L(1)\) |
\(\approx\) |
\(0.4234021087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4234021087\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 7.12T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.082600818101672452751385614439, −8.183843334691155677011810559289, −7.64469911968013578429522544289, −6.77752908397044180608403870620, −5.74350005772684411669623741412, −4.88904497444920443529843668000, −4.40713405169146397278875608132, −3.59158761092533221702629739948, −2.56352804421660676692055692579, −0.37940921174198785799859144063,
0.37940921174198785799859144063, 2.56352804421660676692055692579, 3.59158761092533221702629739948, 4.40713405169146397278875608132, 4.88904497444920443529843668000, 5.74350005772684411669623741412, 6.77752908397044180608403870620, 7.64469911968013578429522544289, 8.183843334691155677011810559289, 9.082600818101672452751385614439
