L(s) = 1 | + 3.40·3-s − 1.83·5-s − 0.970·7-s + 8.58·9-s + 3.13·11-s + 6.37·13-s − 6.24·15-s + 5.07·17-s − 8.17·19-s − 3.30·21-s + 7.55·23-s − 1.63·25-s + 19.0·27-s − 1.60·29-s − 1.14·31-s + 10.6·33-s + 1.78·35-s − 3.51·37-s + 21.7·39-s + 8.67·41-s − 6.25·43-s − 15.7·45-s − 3.02·47-s − 6.05·49-s + 17.2·51-s − 13.9·53-s − 5.74·55-s + ⋯ |
L(s) = 1 | + 1.96·3-s − 0.820·5-s − 0.366·7-s + 2.86·9-s + 0.944·11-s + 1.76·13-s − 1.61·15-s + 1.23·17-s − 1.87·19-s − 0.720·21-s + 1.57·23-s − 0.326·25-s + 3.65·27-s − 0.298·29-s − 0.205·31-s + 1.85·33-s + 0.301·35-s − 0.577·37-s + 3.47·39-s + 1.35·41-s − 0.954·43-s − 2.34·45-s − 0.441·47-s − 0.865·49-s + 2.42·51-s − 1.91·53-s − 0.774·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.422803715\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.422803715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 + 0.970T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 6.37T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 + 3.51T + 37T^{2} \) |
| 41 | \( 1 - 8.67T + 41T^{2} \) |
| 43 | \( 1 + 6.25T + 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 4.93T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 9.76T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.046927313617763824967506137162, −7.77292850514589039340383640274, −6.66969755873321154446289421006, −6.39629792908616543751491582281, −4.87967216802649062309961749868, −3.90375029658441752976404223118, −3.66481251437776392282705150220, −3.07278737223760568714167404761, −1.90251289719132984166403233311, −1.10787907695777920621391368162,
1.10787907695777920621391368162, 1.90251289719132984166403233311, 3.07278737223760568714167404761, 3.66481251437776392282705150220, 3.90375029658441752976404223118, 4.87967216802649062309961749868, 6.39629792908616543751491582281, 6.66969755873321154446289421006, 7.77292850514589039340383640274, 8.046927313617763824967506137162