| L(s) = 1 | + 2-s + 0.835·3-s + 4-s + 3.89·5-s + 0.835·6-s + 4.48·7-s + 8-s − 2.30·9-s + 3.89·10-s + 0.934·11-s + 0.835·12-s − 2.76·13-s + 4.48·14-s + 3.25·15-s + 16-s − 5.56·17-s − 2.30·18-s + 0.943·19-s + 3.89·20-s + 3.74·21-s + 0.934·22-s + 2.52·23-s + 0.835·24-s + 10.1·25-s − 2.76·26-s − 4.42·27-s + 4.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.482·3-s + 0.5·4-s + 1.74·5-s + 0.340·6-s + 1.69·7-s + 0.353·8-s − 0.767·9-s + 1.23·10-s + 0.281·11-s + 0.241·12-s − 0.767·13-s + 1.19·14-s + 0.839·15-s + 0.250·16-s − 1.34·17-s − 0.542·18-s + 0.216·19-s + 0.870·20-s + 0.816·21-s + 0.199·22-s + 0.527·23-s + 0.170·24-s + 2.03·25-s − 0.542·26-s − 0.852·27-s + 0.847·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.646773882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.646773882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 0.835T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 - 0.934T + 11T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 - 0.943T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 0.962T + 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 - 9.42T + 37T^{2} \) |
| 41 | \( 1 - 0.279T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 7.87T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 9.24T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 2.36T + 97T^{2} \) |
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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
−8.565372276253641855575768113628, −7.67161274849454748434761557678, −6.87906484906095567022858894276, −6.04260052108518340013626688984, −5.31128357247208003701101178887, −4.93218741442237001262028550461, −3.96540426559679976537582481374, −2.52800168212424836682082595799, −2.29933422660268348211691113912, −1.37817003796487803525057261473,
1.37817003796487803525057261473, 2.29933422660268348211691113912, 2.52800168212424836682082595799, 3.96540426559679976537582481374, 4.93218741442237001262028550461, 5.31128357247208003701101178887, 6.04260052108518340013626688984, 6.87906484906095567022858894276, 7.67161274849454748434761557678, 8.565372276253641855575768113628
