| L(s) = 1 | + 1.84i·5-s + 5.22i·11-s + 4.46i·13-s − 2.93i·17-s + 5.65·19-s + 2.16i·23-s + 1.58·25-s + 5.41·29-s + 9.65·31-s + 1.41·37-s + 4.01i·41-s − 3.06i·43-s − 1.65·47-s + 9.65·53-s − 9.65·55-s + ⋯ |
| L(s) = 1 | + 0.826i·5-s + 1.57i·11-s + 1.23i·13-s − 0.710i·17-s + 1.29·19-s + 0.451i·23-s + 0.317·25-s + 1.00·29-s + 1.73·31-s + 0.232·37-s + 0.626i·41-s − 0.466i·43-s − 0.241·47-s + 1.32·53-s − 1.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.232586715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232586715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.84iT - 5T^{2} \) |
| 11 | \( 1 - 5.22iT - 11T^{2} \) |
| 13 | \( 1 - 4.46iT - 13T^{2} \) |
| 17 | \( 1 + 2.93iT - 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 2.16iT - 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 4.01iT - 41T^{2} \) |
| 43 | \( 1 + 3.06iT - 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 1.39iT - 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 + 6.49iT - 71T^{2} \) |
| 73 | \( 1 - 4.90iT - 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 9.05iT - 89T^{2} \) |
| 97 | \( 1 - 9.23iT - 97T^{2} \) |
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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
−7.83579472252377753158920600119, −7.44137298583665812534927151710, −6.63297071740897950788436102799, −6.42618885852681156658299596305, −5.02820007392449884684184592866, −4.73722040166251005319635602705, −3.76654697562102303034333084778, −2.85442764021979219022153888091, −2.19636398978447243626808199452, −1.12799044488498391171230841721,
0.69672725391247157257379945967, 1.12649419564575085522210429317, 2.71940036482702391433907704295, 3.21269425121853667255651378253, 4.18031370965333016131385011644, 5.01225636591815938654432695677, 5.65023013527957614959048326222, 6.15276009484494465860879470207, 7.07267326750012892728183554112, 8.072792751280945938801552804039
