| L(s) = 1 | − 3-s + 0.452·5-s − 4.42·7-s + 9-s − 4.87·11-s − 13-s − 0.452·15-s + 0.228·17-s − 3.65·19-s + 4.42·21-s − 4.47·23-s − 4.79·25-s − 27-s + 4.76·29-s − 4.42·31-s + 4.87·33-s − 2·35-s − 4.47·37-s + 39-s − 5.23·41-s + 4.22·43-s + 0.452·45-s − 0.400·47-s + 12.5·49-s − 0.228·51-s − 10.7·53-s − 2.20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.202·5-s − 1.67·7-s + 0.333·9-s − 1.46·11-s − 0.277·13-s − 0.116·15-s + 0.0553·17-s − 0.838·19-s + 0.964·21-s − 0.932·23-s − 0.959·25-s − 0.192·27-s + 0.884·29-s − 0.793·31-s + 0.848·33-s − 0.338·35-s − 0.735·37-s + 0.160·39-s − 0.818·41-s + 0.644·43-s + 0.0674·45-s − 0.0584·47-s + 1.79·49-s − 0.0319·51-s − 1.47·53-s − 0.297·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06535491919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06535491919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.452T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 17 | \( 1 - 0.228T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 + 0.400T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.50997016220593637501253555255, −6.93539687682800582565415810622, −6.00999800038083011515510249225, −5.95700423280786916772551290053, −4.98107239398205027032128918118, −4.22403639973337423689812667718, −3.32207934748551871637100275120, −2.67937894482775351605065216586, −1.76779404319948714945414900790, −0.11635259865177838985405255805,
0.11635259865177838985405255805, 1.76779404319948714945414900790, 2.67937894482775351605065216586, 3.32207934748551871637100275120, 4.22403639973337423689812667718, 4.98107239398205027032128918118, 5.95700423280786916772551290053, 6.00999800038083011515510249225, 6.93539687682800582565415810622, 7.50997016220593637501253555255
