| L(s) = 1 | − 0.290·2-s − 1.26·3-s − 1.91·4-s + 3.23·5-s + 0.368·6-s + 1.13·8-s − 1.39·9-s − 0.941·10-s − 3.38·11-s + 2.42·12-s − 2.06·13-s − 4.10·15-s + 3.49·16-s − 1.31·17-s + 0.406·18-s + 3.39·19-s − 6.20·20-s + 0.984·22-s − 2.82·23-s − 1.44·24-s + 5.48·25-s + 0.599·26-s + 5.56·27-s + 4.41·29-s + 1.19·30-s + 5.50·31-s − 3.29·32-s + ⋯ |
| L(s) = 1 | − 0.205·2-s − 0.731·3-s − 0.957·4-s + 1.44·5-s + 0.150·6-s + 0.402·8-s − 0.465·9-s − 0.297·10-s − 1.02·11-s + 0.700·12-s − 0.572·13-s − 1.05·15-s + 0.874·16-s − 0.320·17-s + 0.0957·18-s + 0.777·19-s − 1.38·20-s + 0.209·22-s − 0.588·23-s − 0.294·24-s + 1.09·25-s + 0.117·26-s + 1.07·27-s + 0.819·29-s + 0.217·30-s + 0.989·31-s − 0.582·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9462873776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9462873776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.290T + 2T^{2} \) |
| 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 + 4.83T + 37T^{2} \) |
| 43 | \( 1 - 8.33T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.654T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 7.52T + 71T^{2} \) |
| 73 | \( 1 + 9.83T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 3.04T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−9.307972292995836325709265599312, −8.472440446231156713330923204581, −7.73690310967227470241584719052, −6.54678665269842675748781345478, −5.83558464723432941504930610625, −5.15746450213162587011097066555, −4.71098905782807253984779704629, −3.15031448891340497303869525884, −2.12601390779664402277669555151, −0.68709993255791251859907935652,
0.68709993255791251859907935652, 2.12601390779664402277669555151, 3.15031448891340497303869525884, 4.71098905782807253984779704629, 5.15746450213162587011097066555, 5.83558464723432941504930610625, 6.54678665269842675748781345478, 7.73690310967227470241584719052, 8.472440446231156713330923204581, 9.307972292995836325709265599312
