| L(s) = 1 | + 3-s + 5-s + 2.60·7-s + 9-s − 2.60·11-s + 15-s + 1.60·17-s + 6.60·19-s + 2.60·21-s + 2.60·23-s − 4·25-s + 27-s + 3·29-s − 5.21·31-s − 2.60·33-s + 2.60·35-s + 2.39·37-s + 11.6·41-s − 2.60·43-s + 45-s + 1.39·47-s − 0.211·49-s + 1.60·51-s + 3·53-s − 2.60·55-s + 6.60·57-s − 9.21·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.984·7-s + 0.333·9-s − 0.785·11-s + 0.258·15-s + 0.389·17-s + 1.51·19-s + 0.568·21-s + 0.543·23-s − 0.800·25-s + 0.192·27-s + 0.557·29-s − 0.935·31-s − 0.453·33-s + 0.440·35-s + 0.393·37-s + 1.81·41-s − 0.397·43-s + 0.149·45-s + 0.203·47-s − 0.0301·49-s + 0.224·51-s + 0.412·53-s − 0.351·55-s + 0.874·57-s − 1.19·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.069856076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.069856076\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.60T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 9.39T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.252189527552748129295664608403, −7.77113233235671691382517770559, −7.27609295812217943033153849336, −6.13514196085818447930594990390, −5.33404816422257962715496742794, −4.81950428122322735099341672722, −3.75248119052276745197935949419, −2.85958004112616051843198380385, −2.02406357649678618250455201536, −1.03679981083505634835911507223,
1.03679981083505634835911507223, 2.02406357649678618250455201536, 2.85958004112616051843198380385, 3.75248119052276745197935949419, 4.81950428122322735099341672722, 5.33404816422257962715496742794, 6.13514196085818447930594990390, 7.27609295812217943033153849336, 7.77113233235671691382517770559, 8.252189527552748129295664608403
