L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + 1.61i·7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)12-s + (0.587 + 0.190i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)19-s + (1.30 − 0.951i)21-s + (0.951 − 0.309i)27-s + (0.951 + 1.30i)28-s + (1.30 + 0.951i)31-s + (0.309 + 0.951i)36-s + (−0.587 − 0.190i)37-s + (−0.190 − 0.587i)39-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + 1.61i·7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)12-s + (0.587 + 0.190i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)19-s + (1.30 − 0.951i)21-s + (0.951 − 0.309i)27-s + (0.951 + 1.30i)28-s + (1.30 + 0.951i)31-s + (0.309 + 0.951i)36-s + (−0.587 − 0.190i)37-s + (−0.190 − 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.175294141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175294141\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - 1.61iT - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−9.391829573068222482994469779278, −8.470601193779764056153936188873, −7.80879718033513055432158258237, −6.69892626829503786097652054294, −6.28752317562240330201402042642, −5.52944901579529506284120101342, −4.95189029865496025224413786544, −3.13217450272737320495444632301, −2.24726181138847089410396743827, −1.37882249229111079078397906110,
1.09031273735873768404186523438, 2.83646655369922482286150962358, 3.78868687353925035687395458590, 4.27646716121372417093779211507, 5.42482274342946827408252351234, 6.41937969329933345830321219682, 6.98505175608264517104208257075, 7.78979507560133526041140296702, 8.609937632698669853413229645864, 9.705720838354120001566462437522