| L(s) = 1 | − 2.12·2-s − 0.178·3-s + 2.51·4-s + 3.60·5-s + 0.380·6-s − 1.10·8-s − 2.96·9-s − 7.66·10-s + 3.99·11-s − 0.450·12-s − 0.644·15-s − 2.69·16-s + 4.78·17-s + 6.30·18-s − 3.15·19-s + 9.07·20-s − 8.48·22-s − 2.17·23-s + 0.196·24-s + 8.00·25-s + 1.06·27-s − 6.57·29-s + 1.37·30-s + 1.48·31-s + 7.93·32-s − 0.714·33-s − 10.1·34-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.103·3-s + 1.25·4-s + 1.61·5-s + 0.155·6-s − 0.389·8-s − 0.989·9-s − 2.42·10-s + 1.20·11-s − 0.129·12-s − 0.166·15-s − 0.674·16-s + 1.16·17-s + 1.48·18-s − 0.722·19-s + 2.03·20-s − 1.80·22-s − 0.454·23-s + 0.0401·24-s + 1.60·25-s + 0.205·27-s − 1.22·29-s + 0.250·30-s + 0.267·31-s + 1.40·32-s − 0.124·33-s − 1.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256160098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256160098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 3 | \( 1 + 0.178T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 - 1.48T + 31T^{2} \) |
| 37 | \( 1 - 4.96T + 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 1.01T + 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 + 8.80T + 71T^{2} \) |
| 73 | \( 1 - 3.08T + 73T^{2} \) |
| 79 | \( 1 - 1.96T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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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
−8.006703405265088134632189349876, −7.21837214986659584064457498058, −6.41517616851967717185012136853, −5.98578124206762981071633271917, −5.37086640784502704994035686155, −4.27721001039062469990264858489, −3.15477274459670530104356967745, −2.20179162429751790609762173192, −1.63197488787468653616778528657, −0.72257869996768103727625394161,
0.72257869996768103727625394161, 1.63197488787468653616778528657, 2.20179162429751790609762173192, 3.15477274459670530104356967745, 4.27721001039062469990264858489, 5.37086640784502704994035686155, 5.98578124206762981071633271917, 6.41517616851967717185012136853, 7.21837214986659584064457498058, 8.006703405265088134632189349876
