L(s) = 1 | + (0.587 + 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 + 1.53i)4-s + (−0.587 + 0.809i)5-s + (1.53 + 1.11i)6-s + (−2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + (−1.80 − 0.587i)10-s + (0.363 + 1.11i)11-s + (−0.809 + 2.48i)12-s + (−0.309 + 0.951i)13-s + i·15-s + (1.00 − 3.07i)16-s + 1.90·18-s − 2.61i·20-s + ⋯ |
L(s) = 1 | + (0.587 + 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 + 1.53i)4-s + (−0.587 + 0.809i)5-s + (1.53 + 1.11i)6-s + (−2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + (−1.80 − 0.587i)10-s + (0.363 + 1.11i)11-s + (−0.809 + 2.48i)12-s + (−0.309 + 0.951i)13-s + i·15-s + (1.00 − 3.07i)16-s + 1.90·18-s − 2.61i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.339609125\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.339609125\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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
−10.28849300206602099080090523448, −9.179621321939717264067485931416, −8.718858388928669773986927540546, −7.53129406092099468457690454020, −7.27057356003313612541406648687, −6.70708443816850815922507759894, −5.72407382494152998444151979303, −4.28439613837111416299783499174, −3.90642016622763344140015691888, −2.49157129068682076206630440115,
1.08636891455385831220793980441, 2.59710854173168668276687508654, 3.40936255059050582905613116552, 4.14666798425654193608065860955, 4.95655533511474684822948974713, 5.79608457158030004233991617165, 7.83989864870606852157745498066, 8.538211842490619156133781013162, 9.278907079430958140517446404886, 9.863168208952986035369697247096