L(s) = 1 | + 2.46·2-s + 4.07·4-s + 0.653·5-s + 7-s + 5.11·8-s + 1.61·10-s + 2.42·11-s + 4.26·13-s + 2.46·14-s + 4.46·16-s − 2.38·17-s − 5.35·19-s + 2.66·20-s + 5.97·22-s + 23-s − 4.57·25-s + 10.5·26-s + 4.07·28-s − 3.23·29-s + 0.388·31-s + 0.769·32-s − 5.88·34-s + 0.653·35-s + 9.31·37-s − 13.1·38-s + 3.34·40-s − 5.73·41-s + ⋯ |
L(s) = 1 | + 1.74·2-s + 2.03·4-s + 0.292·5-s + 0.377·7-s + 1.80·8-s + 0.509·10-s + 0.730·11-s + 1.18·13-s + 0.658·14-s + 1.11·16-s − 0.579·17-s − 1.22·19-s + 0.595·20-s + 1.27·22-s + 0.208·23-s − 0.914·25-s + 2.06·26-s + 0.770·28-s − 0.600·29-s + 0.0697·31-s + 0.135·32-s − 1.00·34-s + 0.110·35-s + 1.53·37-s − 2.14·38-s + 0.529·40-s − 0.894·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.342691851\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.342691851\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 - 0.653T + 5T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 0.388T + 31T^{2} \) |
| 37 | \( 1 - 9.31T + 37T^{2} \) |
| 41 | \( 1 + 5.73T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 1.18T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 2.00T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 + 6.07T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 - 0.0112T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.543153161828208106169949830004, −8.665285085281746098245355966723, −7.68071878152401801810866345143, −6.54188593670510091225262653045, −6.18857255069453120421485758447, −5.33389264968437426012314506799, −4.24006939781376570562638045585, −3.88364053689209564595947871843, −2.60555125049593158330926400402, −1.61667524175534746634977226468,
1.61667524175534746634977226468, 2.60555125049593158330926400402, 3.88364053689209564595947871843, 4.24006939781376570562638045585, 5.33389264968437426012314506799, 6.18857255069453120421485758447, 6.54188593670510091225262653045, 7.68071878152401801810866345143, 8.665285085281746098245355966723, 9.543153161828208106169949830004