| L(s) = 1 | + 3·3-s − 5·5-s + 12.8·7-s + 9·9-s + 49.7·11-s + 52.6·13-s − 15·15-s + 84.6·17-s − 26.6·19-s + 38.6·21-s − 136.·23-s + 25·25-s + 27·27-s − 6·29-s − 47.1·31-s + 149.·33-s − 64.3·35-s − 344.·37-s + 157.·39-s − 43.2·41-s + 252·43-s − 45·45-s − 306.·47-s − 177.·49-s + 253.·51-s + 455.·53-s − 248.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.694·7-s + 0.333·9-s + 1.36·11-s + 1.12·13-s − 0.258·15-s + 1.20·17-s − 0.321·19-s + 0.401·21-s − 1.23·23-s + 0.200·25-s + 0.192·27-s − 0.0384·29-s − 0.273·31-s + 0.787·33-s − 0.310·35-s − 1.52·37-s + 0.647·39-s − 0.164·41-s + 0.893·43-s − 0.149·45-s − 0.949·47-s − 0.517·49-s + 0.696·51-s + 1.17·53-s − 0.609·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.142701512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.142701512\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 12.8T + 343T^{2} \) |
| 11 | \( 1 - 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 43.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252T + 7.95e4T^{2} \) |
| 47 | \( 1 + 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 455.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 708.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 652.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 704.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 531.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 57.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 429.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 227.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 152.T + 9.12e5T^{2} \) |
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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
−9.570700777087837403105439394017, −8.570282874776049121251227720784, −8.217707404741600505929734255393, −7.20266819913761144000692791633, −6.30992905251377243628461490650, −5.24619901587105509511186047975, −3.93585878879025045034605611458, −3.60491304426032248986770349818, −1.97286516672397735495687523576, −1.00794319535921415067595237354,
1.00794319535921415067595237354, 1.97286516672397735495687523576, 3.60491304426032248986770349818, 3.93585878879025045034605611458, 5.24619901587105509511186047975, 6.30992905251377243628461490650, 7.20266819913761144000692791633, 8.217707404741600505929734255393, 8.570282874776049121251227720784, 9.570700777087837403105439394017
