| L(s) = 1 | + 0.278·2-s − 3-s − 1.92·4-s − 3.63·5-s − 0.278·6-s + 0.219·7-s − 1.09·8-s + 9-s − 1.01·10-s − 1.42·11-s + 1.92·12-s − 2.01·13-s + 0.0610·14-s + 3.63·15-s + 3.54·16-s + 17-s + 0.278·18-s − 4.98·19-s + 6.99·20-s − 0.219·21-s − 0.397·22-s − 8.82·23-s + 1.09·24-s + 8.22·25-s − 0.559·26-s − 27-s − 0.421·28-s + ⋯ |
| L(s) = 1 | + 0.196·2-s − 0.577·3-s − 0.961·4-s − 1.62·5-s − 0.113·6-s + 0.0829·7-s − 0.385·8-s + 0.333·9-s − 0.319·10-s − 0.430·11-s + 0.554·12-s − 0.557·13-s + 0.0163·14-s + 0.938·15-s + 0.885·16-s + 0.242·17-s + 0.0655·18-s − 1.14·19-s + 1.56·20-s − 0.0478·21-s − 0.0846·22-s − 1.83·23-s + 0.222·24-s + 1.64·25-s − 0.109·26-s − 0.192·27-s − 0.0797·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 0.278T + 2T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 - 0.219T + 7T^{2} \) |
| 11 | \( 1 + 1.42T + 11T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 - 8.45T + 31T^{2} \) |
| 37 | \( 1 - 0.132T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 7.42T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.67T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
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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
−7.78925459676022293377365592279, −6.71861345922753231912773920463, −6.12141198520205466403749404104, −5.11815948097915040702668033233, −4.61883028341908181575788455658, −4.05679668674956391554514805031, −3.45213939135027444803868030426, −2.32722317764523006069035311982, −0.76081748007591137761058135730, 0,
0.76081748007591137761058135730, 2.32722317764523006069035311982, 3.45213939135027444803868030426, 4.05679668674956391554514805031, 4.61883028341908181575788455658, 5.11815948097915040702668033233, 6.12141198520205466403749404104, 6.71861345922753231912773920463, 7.78925459676022293377365592279
