L(s) = 1 | + 2.00·2-s + 2.03·4-s − 0.00545·5-s − 3.82·7-s + 0.0695·8-s − 0.0109·10-s − 3.53·13-s − 7.68·14-s − 3.92·16-s + 4.57·17-s − 0.684·19-s − 0.0110·20-s + 1.52·23-s − 4.99·25-s − 7.09·26-s − 7.78·28-s − 0.362·29-s + 4.28·31-s − 8.03·32-s + 9.18·34-s + 0.0208·35-s + 6.16·37-s − 1.37·38-s − 0.000379·40-s + 6.67·41-s + 7.31·43-s + 3.05·46-s + ⋯ |
L(s) = 1 | + 1.42·2-s + 1.01·4-s − 0.00243·5-s − 1.44·7-s + 0.0245·8-s − 0.00346·10-s − 0.980·13-s − 2.05·14-s − 0.982·16-s + 1.10·17-s − 0.157·19-s − 0.00248·20-s + 0.317·23-s − 0.999·25-s − 1.39·26-s − 1.47·28-s − 0.0673·29-s + 0.770·31-s − 1.41·32-s + 1.57·34-s + 0.00352·35-s + 1.01·37-s − 0.223·38-s − 5.99e − 5·40-s + 1.04·41-s + 1.11·43-s + 0.450·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.878398976\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.878398976\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 5 | \( 1 + 0.00545T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 + 0.684T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 0.362T + 29T^{2} \) |
| 31 | \( 1 - 4.28T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 6.67T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 2.78T + 71T^{2} \) |
| 73 | \( 1 + 0.0668T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 - 0.897T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.41472411830774317881141043516, −6.73917283072953506281251873857, −6.10562686464962916000696424536, −5.66656580022969438495892228942, −4.87182221798370994104410934358, −4.16819531086776355371712058784, −3.46271465507936385334118354088, −2.89709498096031253542130712250, −2.21339351800525989277827704523, −0.61235492654099851037797466561,
0.61235492654099851037797466561, 2.21339351800525989277827704523, 2.89709498096031253542130712250, 3.46271465507936385334118354088, 4.16819531086776355371712058784, 4.87182221798370994104410934358, 5.66656580022969438495892228942, 6.10562686464962916000696424536, 6.73917283072953506281251873857, 7.41472411830774317881141043516