| L(s) = 1 | + 2.43·3-s + 1.30·5-s + 7-s + 2.92·9-s − 11-s − 13-s + 3.18·15-s + 5.30·17-s − 1.44·19-s + 2.43·21-s + 8.93·23-s − 3.28·25-s − 0.181·27-s − 5.58·29-s − 6.13·31-s − 2.43·33-s + 1.30·35-s + 8.97·37-s − 2.43·39-s − 3.64·41-s + 6.67·43-s + 3.83·45-s − 2.40·47-s + 49-s + 12.9·51-s + 4.61·53-s − 1.30·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 0.585·5-s + 0.377·7-s + 0.975·9-s − 0.301·11-s − 0.277·13-s + 0.823·15-s + 1.28·17-s − 0.331·19-s + 0.531·21-s + 1.86·23-s − 0.657·25-s − 0.0349·27-s − 1.03·29-s − 1.10·31-s − 0.423·33-s + 0.221·35-s + 1.47·37-s − 0.389·39-s − 0.569·41-s + 1.01·43-s + 0.571·45-s − 0.351·47-s + 0.142·49-s + 1.80·51-s + 0.634·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.349822959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.349822959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 - 8.93T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 - 8.97T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 - 6.67T + 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.118T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−7.73801466074942099998853064564, −7.49479718873950888148563518238, −6.60376652140844598259221341988, −5.51534272022517009038467995115, −5.21263282636098172801976288484, −4.00815060493492054520001230669, −3.46053202266129969278695975108, −2.55417936202686031582718895452, −2.05997796194135093907712092708, −0.997931210659447750062113828744,
0.997931210659447750062113828744, 2.05997796194135093907712092708, 2.55417936202686031582718895452, 3.46053202266129969278695975108, 4.00815060493492054520001230669, 5.21263282636098172801976288484, 5.51534272022517009038467995115, 6.60376652140844598259221341988, 7.49479718873950888148563518238, 7.73801466074942099998853064564
