L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.309 − 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)7-s + (−0.309 + 0.951i)9-s + (0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.363 − 0.5i)13-s − 15-s + (−0.809 − 0.587i)16-s + (0.951 + 0.690i)17-s + (−0.587 − 0.809i)20-s + (0.309 + 0.951i)21-s + (−0.309 − 0.951i)25-s + (0.951 − 0.309i)27-s + (−0.587 + 0.809i)28-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.309 − 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)7-s + (−0.309 + 0.951i)9-s + (0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.363 − 0.5i)13-s − 15-s + (−0.809 − 0.587i)16-s + (0.951 + 0.690i)17-s + (−0.587 − 0.809i)20-s + (0.309 + 0.951i)21-s + (−0.309 − 0.951i)25-s + (0.951 − 0.309i)27-s + (−0.587 + 0.809i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8965683037\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8965683037\) |
\(L(1)\) |
|
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 | \( 1 + (-0.587 + 0.809i)T \) |
| 7 | \( 1 + (0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
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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
−9.761085017462901277048154969510, −9.005506030792556949439757150119, −7.956215127767609554798944086352, −6.88961153795617002771612454202, −6.22569441396110613870537846069, −5.65558200291929615643508290232, −4.86859738626285699447213127543, −3.30205462792196647919216545507, −1.81725985449129133163190639007, −0.883182437990402172295430772671,
2.31535240084069125657915237706, 3.31784775946854281687624922032, 4.01572193588355142139484238141, 5.25918221459381387521170654537, 6.35064742725919505558790733772, 6.73528243182721311513669069515, 7.67650558217900215721170146888, 9.008472707855024382793025604575, 9.687371003254752550258689224818, 10.03819528264774275741600589637