| L(s) = 1 | − 0.618·3-s + 0.381·5-s − 2.61·9-s − 4.47·11-s + 4.47·13-s − 0.236·15-s − 17-s − 2.47·19-s − 9.23·23-s − 4.85·25-s + 3.47·27-s + 2·29-s − 2.09·31-s + 2.76·33-s + 4·37-s − 2.76·39-s − 1.09·41-s + 1.09·43-s − 45-s + 6.76·47-s + 0.618·51-s + 8.09·53-s − 1.70·55-s + 1.52·57-s + 4·59-s − 5.38·61-s + 1.70·65-s + ⋯ |
| L(s) = 1 | − 0.356·3-s + 0.170·5-s − 0.872·9-s − 1.34·11-s + 1.24·13-s − 0.0609·15-s − 0.242·17-s − 0.567·19-s − 1.92·23-s − 0.970·25-s + 0.668·27-s + 0.371·29-s − 0.375·31-s + 0.481·33-s + 0.657·37-s − 0.442·39-s − 0.170·41-s + 0.166·43-s − 0.149·45-s + 0.986·47-s + 0.0865·51-s + 1.11·53-s − 0.230·55-s + 0.202·57-s + 0.520·59-s − 0.689·61-s + 0.211·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013446904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013446904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 9.23T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 8.09T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 0.0901T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.194236498817811408310884688340, −7.36171742474818568335847441292, −6.30169116369892121265963741874, −5.89111199141954782635656305458, −5.38450113866789782789353591690, −4.35958957616347710224273411126, −3.65502559918208405714508557069, −2.63054236298960805733130400838, −1.94919612182651566776652957879, −0.50187999603879747533974664013,
0.50187999603879747533974664013, 1.94919612182651566776652957879, 2.63054236298960805733130400838, 3.65502559918208405714508557069, 4.35958957616347710224273411126, 5.38450113866789782789353591690, 5.89111199141954782635656305458, 6.30169116369892121265963741874, 7.36171742474818568335847441292, 8.194236498817811408310884688340
