| L(s) = 1 | + 2.66i·5-s − 2.66i·7-s + 11-s + 3.60·13-s − 4.66i·17-s − 3.04i·19-s − 0.476·23-s − 2.12·25-s + 3.46i·29-s + 1.19i·31-s + 7.12·35-s + 8.37·37-s + 1.19i·41-s − 10.7i·43-s − 8.76·47-s + ⋯ |
| L(s) = 1 | + 1.19i·5-s − 1.00i·7-s + 0.301·11-s + 0.998·13-s − 1.13i·17-s − 0.698i·19-s − 0.0994·23-s − 0.424·25-s + 0.643i·29-s + 0.215i·31-s + 1.20·35-s + 1.37·37-s + 0.187i·41-s − 1.64i·43-s − 1.27·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.851600512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851600512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.66iT - 5T^{2} \) |
| 7 | \( 1 + 2.66iT - 7T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 4.66iT - 17T^{2} \) |
| 19 | \( 1 + 3.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.476T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 1.19iT - 31T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 - 1.19iT - 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 - 4.25iT - 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 0.398T + 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 0.476T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 + 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−7.56647164620786761491491915164, −7.31444205185822170855835779620, −6.55020197181867500669150061253, −6.07899828342311109396079077740, −4.94921101072322881378829254494, −4.23294796518016938641574639204, −3.33889789675306801419316239265, −2.88661795612716088050685268197, −1.65275731034834015410615197178, −0.51468862985453817488257688613,
1.09402804966479524087353108164, 1.77698309385182677774650405321, 2.87957866905549684559885470453, 3.91753699784432398728752762216, 4.47554265421747275371532681384, 5.39250744003506939450727906665, 6.02511328555029443438215346277, 6.39915213223552386349711502668, 7.73665195723265297196930286355, 8.317088878080908210915266674771
