| L(s) = 1 | + 3-s + 1.90·5-s + 2.81·7-s + 9-s + 0.318·11-s + 5.12·13-s + 1.90·15-s + 3.73·17-s − 0.725·19-s + 2.81·21-s + 0.612·23-s − 1.36·25-s + 27-s − 3.87·29-s − 6.65·31-s + 0.318·33-s + 5.36·35-s − 9.04·37-s + 5.12·39-s + 9.55·41-s − 10.9·43-s + 1.90·45-s + 8.76·47-s + 0.915·49-s + 3.73·51-s − 2.46·53-s + 0.608·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.853·5-s + 1.06·7-s + 0.333·9-s + 0.0961·11-s + 1.42·13-s + 0.492·15-s + 0.905·17-s − 0.166·19-s + 0.613·21-s + 0.127·23-s − 0.272·25-s + 0.192·27-s − 0.720·29-s − 1.19·31-s + 0.0555·33-s + 0.907·35-s − 1.48·37-s + 0.820·39-s + 1.49·41-s − 1.66·43-s + 0.284·45-s + 1.27·47-s + 0.130·49-s + 0.522·51-s − 0.338·53-s + 0.0820·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.129704572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.129704572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 0.318T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 + 0.725T + 19T^{2} \) |
| 23 | \( 1 - 0.612T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 - 9.55T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 8.76T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 2.88T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 - 4.04T + 83T^{2} \) |
| 89 | \( 1 + 9.15T + 89T^{2} \) |
| 97 | \( 1 + 9.62T + 97T^{2} \) |
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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
−9.096223810189071602701685002126, −8.411067308664371144096413278598, −7.76505547882764548671163885266, −6.85977350572714556753883204338, −5.81731300193932341364080044348, −5.30481722571042835055111240133, −4.11500765967001792998898293796, −3.32244521001959551990463815789, −2.00131805825005205916599038555, −1.36832628583485687007785330895,
1.36832628583485687007785330895, 2.00131805825005205916599038555, 3.32244521001959551990463815789, 4.11500765967001792998898293796, 5.30481722571042835055111240133, 5.81731300193932341364080044348, 6.85977350572714556753883204338, 7.76505547882764548671163885266, 8.411067308664371144096413278598, 9.096223810189071602701685002126
