L(s) = 1 | + (−0.280 − 0.863i)2-s + (0.142 − 0.103i)4-s + (−0.297 − 0.0966i)5-s + (−0.863 − 0.627i)8-s + 0.284i·10-s + (0.891 − 0.453i)11-s + (0.951 − 0.309i)13-s + (−0.245 + 0.754i)16-s + (−0.0522 + 0.0169i)20-s + (−0.642 − 0.642i)22-s + (−0.729 − 0.530i)25-s + (−0.533 − 0.734i)26-s − 0.346·32-s + (0.196 + 0.270i)40-s + (−1.14 − 0.831i)41-s + ⋯ |
L(s) = 1 | + (−0.280 − 0.863i)2-s + (0.142 − 0.103i)4-s + (−0.297 − 0.0966i)5-s + (−0.863 − 0.627i)8-s + 0.284i·10-s + (0.891 − 0.453i)11-s + (0.951 − 0.309i)13-s + (−0.245 + 0.754i)16-s + (−0.0522 + 0.0169i)20-s + (−0.642 − 0.642i)22-s + (−0.729 − 0.530i)25-s + (−0.533 − 0.734i)26-s − 0.346·32-s + (0.196 + 0.270i)40-s + (−1.14 − 0.831i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9754551906\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9754551906\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.891 + 0.453i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
good | 2 | \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (0.297 + 0.0966i)T + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 1.17iT - T^{2} \) |
| 47 | \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.04 - 1.44i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.0966 + 0.297i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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
−9.698626953263280383204222564121, −8.897598443493836823113532171549, −8.280883142212281308061541813647, −7.07238466390590728951345443452, −6.28401206697511331679176102280, −5.50923697016820290558272063152, −4.01133532205184470111770083322, −3.43111050166300276988529382436, −2.17332434967664430056988954627, −0.987535539520318366802433135556,
1.74106556816453233049171766361, 3.19049690328518243197324266228, 4.09094784393989946124133240960, 5.29331349632604396446919642455, 6.34228156039065224718763724929, 6.73985425704606144174956149033, 7.70288308207463406788603407557, 8.330945468650909492402339283160, 9.139268082763574149576686789193, 9.840732330163134674220509886588