L(s) = 1 | + (0.654 − 0.755i)2-s + (0.909 − 0.584i)3-s + (−0.142 − 0.989i)4-s + (0.234 + 0.512i)5-s + (0.153 − 1.07i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.0702 − 0.153i)9-s + (0.540 + 0.158i)10-s + (−0.708 − 0.817i)12-s + (−1.45 − 0.425i)13-s + (0.415 − 0.909i)14-s + (0.512 + 0.329i)15-s + (−0.959 + 0.281i)16-s + (−0.0702 − 0.153i)18-s + (0.258 + 1.80i)19-s + ⋯ |
L(s) = 1 | + (0.654 − 0.755i)2-s + (0.909 − 0.584i)3-s + (−0.142 − 0.989i)4-s + (0.234 + 0.512i)5-s + (0.153 − 1.07i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.0702 − 0.153i)9-s + (0.540 + 0.158i)10-s + (−0.708 − 0.817i)12-s + (−1.45 − 0.425i)13-s + (0.415 − 0.909i)14-s + (0.512 + 0.329i)15-s + (−0.959 + 0.281i)16-s + (−0.0702 − 0.153i)18-s + (0.258 + 1.80i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.009163087\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.009163087\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.959 + 0.281i)T \) |
good | 3 | \( 1 + (-0.909 + 0.584i)T + (0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (-0.234 - 0.512i)T + (-0.654 + 0.755i)T^{2} \) |
| 11 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (1.45 + 0.425i)T + (0.841 + 0.540i)T^{2} \) |
| 17 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
| 29 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 41 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (1.74 + 0.512i)T + (0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.822 + 1.80i)T + (-0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.654 - 0.755i)T^{2} \) |
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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
−10.02370654987739637252886070921, −8.841016579416964897951312929249, −7.920812874783745611971917120646, −7.40418375774525263589238947922, −6.23976763445782152592492977769, −5.28667947244851088338692061869, −4.38418384034353114018900424567, −3.26248926658923857093930475600, −2.36925230612131970056180828466, −1.64328815021791284846094714116,
2.22514373369760617176568979740, 3.09543914696085992489458450227, 4.40599664824337242826809447314, 4.78800383350275973790338752828, 5.66498524995414722552886873718, 6.89577727376503595331389553203, 7.66851989768373235491724667797, 8.429163140334634526476241604012, 9.203476643098666542480576379063, 9.521397508773570968131440103236