| L(s) = 1 | − 2-s − 2.31·3-s + 4-s − 1.83·5-s + 2.31·6-s + 0.291·7-s − 8-s + 2.34·9-s + 1.83·10-s − 0.391·11-s − 2.31·12-s + 0.600·13-s − 0.291·14-s + 4.24·15-s + 16-s + 8.06·17-s − 2.34·18-s − 1.02·19-s − 1.83·20-s − 0.674·21-s + 0.391·22-s + 23-s + 2.31·24-s − 1.62·25-s − 0.600·26-s + 1.51·27-s + 0.291·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.33·3-s + 0.5·4-s − 0.821·5-s + 0.943·6-s + 0.110·7-s − 0.353·8-s + 0.781·9-s + 0.581·10-s − 0.118·11-s − 0.667·12-s + 0.166·13-s − 0.0780·14-s + 1.09·15-s + 0.250·16-s + 1.95·17-s − 0.552·18-s − 0.235·19-s − 0.410·20-s − 0.147·21-s + 0.0834·22-s + 0.208·23-s + 0.471·24-s − 0.324·25-s − 0.117·26-s + 0.291·27-s + 0.0551·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6723441486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6723441486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 - 0.291T + 7T^{2} \) |
| 11 | \( 1 + 0.391T + 11T^{2} \) |
| 13 | \( 1 - 0.600T + 13T^{2} \) |
| 17 | \( 1 - 8.06T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 29 | \( 1 - 6.44T + 29T^{2} \) |
| 31 | \( 1 - 9.49T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 + 1.05T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 - 9.67T + 59T^{2} \) |
| 61 | \( 1 - 0.715T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.39T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 3.64T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
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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
−8.182497371426941223632343868627, −7.41464578822887882790405800065, −6.60543677101565665997815485135, −6.10044214578518966951293087232, −5.26449380064527407588664941318, −4.65621667306430566127319348197, −3.62473859599866095172219102942, −2.78352063004734460210497534274, −1.30980032195705009027151295843, −0.58349738845528266452422001840,
0.58349738845528266452422001840, 1.30980032195705009027151295843, 2.78352063004734460210497534274, 3.62473859599866095172219102942, 4.65621667306430566127319348197, 5.26449380064527407588664941318, 6.10044214578518966951293087232, 6.60543677101565665997815485135, 7.41464578822887882790405800065, 8.182497371426941223632343868627
