L(s) = 1 | + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.948 − 1.73i)5-s + (−0.281 + 0.959i)8-s + (1.58 + 1.18i)10-s + (−0.769 + 1.53i)13-s + (−0.142 − 0.989i)16-s + (−0.398 + 0.148i)17-s + (−1.93 − 0.420i)20-s + (−1.57 + 2.45i)25-s + (0.0613 − 1.71i)26-s + (0.175 + 1.63i)29-s + (0.540 + 0.841i)32-s + (0.300 − 0.300i)34-s + (0.0659 − 0.0273i)37-s + ⋯ |
L(s) = 1 | + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.948 − 1.73i)5-s + (−0.281 + 0.959i)8-s + (1.58 + 1.18i)10-s + (−0.769 + 1.53i)13-s + (−0.142 − 0.989i)16-s + (−0.398 + 0.148i)17-s + (−1.93 − 0.420i)20-s + (−1.57 + 2.45i)25-s + (0.0613 − 1.71i)26-s + (0.175 + 1.63i)29-s + (0.540 + 0.841i)32-s + (0.300 − 0.300i)34-s + (0.0659 − 0.0273i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3383378551\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3383378551\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.909 - 0.415i)T \) |
| 3 | \( 1 \) |
| 89 | \( 1 + (0.877 + 0.479i)T \) |
good | 5 | \( 1 + (0.948 + 1.73i)T + (-0.540 + 0.841i)T^{2} \) |
| 7 | \( 1 + (0.212 + 0.977i)T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.769 - 1.53i)T + (-0.599 - 0.800i)T^{2} \) |
| 17 | \( 1 + (0.398 - 0.148i)T + (0.755 - 0.654i)T^{2} \) |
| 19 | \( 1 + (0.877 - 0.479i)T^{2} \) |
| 23 | \( 1 + (0.479 + 0.877i)T^{2} \) |
| 29 | \( 1 + (-0.175 - 1.63i)T + (-0.977 + 0.212i)T^{2} \) |
| 31 | \( 1 + (-0.479 + 0.877i)T^{2} \) |
| 37 | \( 1 + (-0.0659 + 0.0273i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.400 + 0.800i)T + (-0.599 + 0.800i)T^{2} \) |
| 43 | \( 1 + (-0.977 - 0.212i)T^{2} \) |
| 47 | \( 1 + (-0.989 + 0.142i)T^{2} \) |
| 53 | \( 1 + (0.133 - 1.86i)T + (-0.989 - 0.142i)T^{2} \) |
| 59 | \( 1 + (-0.800 - 0.599i)T^{2} \) |
| 61 | \( 1 + (1.23 - 0.222i)T + (0.936 - 0.349i)T^{2} \) |
| 67 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 73 | \( 1 + (-1.29 + 0.186i)T + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 83 | \( 1 + (0.0713 - 0.997i)T^{2} \) |
| 97 | \( 1 + (-1.91 - 0.562i)T + (0.841 + 0.540i)T^{2} \) |
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
−9.079935103923378440947013522153, −8.457317715751656422781985686694, −7.58881361916844093922047655242, −7.11396037247816831733227864674, −6.14699679410003416361701756785, −5.02728897385685894250991740882, −4.72072617613003724541588279715, −3.67131824582080156285785076252, −2.08233081754813345799929464247, −1.17716716340565036932901482512,
0.29879257259264496760134103691, 2.25227152926214226944138729871, 2.93903444322738039012534116400, 3.52163764137533449165340328995, 4.54455454291613152538618342830, 5.99787717178848142366911660071, 6.66150651924731933831641733598, 7.42953307680060592131877033899, 7.87107672946513347769211140149, 8.411085775169118753856341082316