| L(s) = 1 | + 3-s − 2.82·5-s − 2.82·7-s + 9-s + 4·11-s + 5.65·13-s − 2.82·15-s − 2·17-s + 4·19-s − 2.82·21-s + 5.65·23-s + 3.00·25-s + 27-s + 2.82·29-s + 8.48·31-s + 4·33-s + 8.00·35-s + 5.65·39-s − 10·41-s + 12·43-s − 2.82·45-s − 5.65·47-s + 1.00·49-s − 2·51-s + 2.82·53-s − 11.3·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.26·5-s − 1.06·7-s + 0.333·9-s + 1.20·11-s + 1.56·13-s − 0.730·15-s − 0.485·17-s + 0.917·19-s − 0.617·21-s + 1.17·23-s + 0.600·25-s + 0.192·27-s + 0.525·29-s + 1.52·31-s + 0.696·33-s + 1.35·35-s + 0.905·39-s − 1.56·41-s + 1.82·43-s − 0.421·45-s − 0.825·47-s + 0.142·49-s − 0.280·51-s + 0.388·53-s − 1.52·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527690048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527690048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.28009841344593893076259941434, −9.212078102163279702311731705058, −8.731262824133501400138219398389, −7.81343294487424868030858234002, −6.82565623673539050240333032784, −6.22865905640795630079773976967, −4.56637920958706365069482256691, −3.64457877857498921465958647906, −3.11104644282753822814754116264, −1.06157146329068855737319521970,
1.06157146329068855737319521970, 3.11104644282753822814754116264, 3.64457877857498921465958647906, 4.56637920958706365069482256691, 6.22865905640795630079773976967, 6.82565623673539050240333032784, 7.81343294487424868030858234002, 8.731262824133501400138219398389, 9.212078102163279702311731705058, 10.28009841344593893076259941434
