L(s) = 1 | + 1.49·2-s + 0.244·4-s + 5-s − 3.23·7-s − 2.62·8-s + 1.49·10-s + 2.15·13-s − 4.84·14-s − 4.42·16-s + 2.54·17-s + 2·19-s + 0.244·20-s − 2.98·23-s + 25-s + 3.23·26-s − 0.791·28-s − 1.21·29-s + 8.46·31-s − 1.37·32-s + 3.81·34-s − 3.23·35-s − 0.546·37-s + 2.99·38-s − 2.62·40-s − 2.84·41-s + 6.19·43-s − 4.46·46-s + ⋯ |
L(s) = 1 | + 1.05·2-s + 0.122·4-s + 0.447·5-s − 1.22·7-s − 0.929·8-s + 0.473·10-s + 0.598·13-s − 1.29·14-s − 1.10·16-s + 0.617·17-s + 0.458·19-s + 0.0547·20-s − 0.621·23-s + 0.200·25-s + 0.633·26-s − 0.149·28-s − 0.225·29-s + 1.51·31-s − 0.243·32-s + 0.654·34-s − 0.546·35-s − 0.0898·37-s + 0.486·38-s − 0.415·40-s − 0.444·41-s + 0.945·43-s − 0.658·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.639108386\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.639108386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.98T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 8.46T + 31T^{2} \) |
| 37 | \( 1 + 0.546T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 - 5.79T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 8.06T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 97 | \( 1 - 3.30T + 97T^{2} \) |
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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
−8.600903519834107751947964698615, −7.54602519340951177225256487417, −6.64939713333506286036478968165, −6.01242782353081023462951735588, −5.61923469911422593937444400396, −4.63513056484257607027285957272, −3.79662616514103077687082426501, −3.19244340920515616590076105107, −2.37307539256546262589194062558, −0.78583423156555059230673448507,
0.78583423156555059230673448507, 2.37307539256546262589194062558, 3.19244340920515616590076105107, 3.79662616514103077687082426501, 4.63513056484257607027285957272, 5.61923469911422593937444400396, 6.01242782353081023462951735588, 6.64939713333506286036478968165, 7.54602519340951177225256487417, 8.600903519834107751947964698615