| L(s) = 1 | − 2-s + 4-s + 3.41·5-s + 0.828·7-s − 8-s − 3.41·10-s + 4·11-s − 2.82·13-s − 0.828·14-s + 16-s − 1.17·19-s + 3.41·20-s − 4·22-s + 8.82·23-s + 6.65·25-s + 2.82·26-s + 0.828·28-s − 3.41·29-s − 4.82·31-s − 32-s + 2.82·35-s + 6.24·37-s + 1.17·38-s − 3.41·40-s + 11.0·41-s − 9.65·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.52·5-s + 0.313·7-s − 0.353·8-s − 1.07·10-s + 1.20·11-s − 0.784·13-s − 0.221·14-s + 0.250·16-s − 0.268·19-s + 0.763·20-s − 0.852·22-s + 1.84·23-s + 1.33·25-s + 0.554·26-s + 0.156·28-s − 0.634·29-s − 0.867·31-s − 0.176·32-s + 0.478·35-s + 1.02·37-s + 0.190·38-s − 0.539·40-s + 1.72·41-s − 1.47·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.229111594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.229111594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.382854857037056308617255635208, −7.41699348368718585520898299135, −6.77206623331877112147368855845, −6.21807837433529896136866233967, −5.39540239644490210449160181563, −4.74034108022596321063241818176, −3.53585261514464642670460975142, −2.49638652474038117486502150350, −1.80759303737366475634753876730, −0.952851822784814163942094901685,
0.952851822784814163942094901685, 1.80759303737366475634753876730, 2.49638652474038117486502150350, 3.53585261514464642670460975142, 4.74034108022596321063241818176, 5.39540239644490210449160181563, 6.21807837433529896136866233967, 6.77206623331877112147368855845, 7.41699348368718585520898299135, 8.382854857037056308617255635208
