L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.366 − 1.36i)5-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (0.366 − 1.36i)10-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + (0.999 − 0.999i)20-s + (−0.866 + 0.5i)25-s + (1 − i)29-s + (−0.866 + 0.499i)32-s + 0.999i·34-s + 0.999i·36-s + (−0.366 − 1.36i)37-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.366 − 1.36i)5-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (0.366 − 1.36i)10-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + (0.999 − 0.999i)20-s + (−0.866 + 0.5i)25-s + (1 − i)29-s + (−0.866 + 0.499i)32-s + 0.999i·34-s + 0.999i·36-s + (−0.366 − 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.167208870\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167208870\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−8.569384817439039317195164175860, −8.017098079554672650007511218787, −7.48571949920054147606610151765, −6.52235462811834535668539023847, −5.69567098547523785107523463382, −4.98403223714129092548207534984, −4.30681260624239830956482994031, −3.80382516343326778635040671762, −2.45050393713236533417445482065, −1.31662503215483504219125268092,
1.23450432191270852472600508450, 2.55805414342024236826004287282, 3.19403757196379110927894446113, 3.91879664829420711408518867411, 4.76371918689662280376998643810, 5.67743266210846920284039915086, 6.67255441168520991732427837074, 6.94634457921484203527128021094, 7.65550748356675501277473555452, 8.886725098303598529344154030287