L(s) = 1 | + (1.08 − 0.909i)2-s + (1.32 + 0.483i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (1.87 − 0.684i)6-s + (0.766 + 0.642i)9-s + (−1.08 − 0.909i)10-s + (0.707 − 1.22i)12-s + (−1.32 + 0.483i)13-s + (0.245 − 1.39i)15-s + (0.939 + 0.342i)16-s + 1.41·18-s − 1.00·20-s + (−0.939 + 0.342i)25-s + (−0.999 + 1.73i)26-s + ⋯ |
L(s) = 1 | + (1.08 − 0.909i)2-s + (1.32 + 0.483i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (1.87 − 0.684i)6-s + (0.766 + 0.642i)9-s + (−1.08 − 0.909i)10-s + (0.707 − 1.22i)12-s + (−1.32 + 0.483i)13-s + (0.245 − 1.39i)15-s + (0.939 + 0.342i)16-s + 1.41·18-s − 1.00·20-s + (−0.939 + 0.342i)25-s + (−0.999 + 1.73i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.813726929\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.813726929\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-1.08 + 0.909i)T + (0.173 - 0.984i)T^{2} \) |
| 3 | \( 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.32 - 0.483i)T + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.08 - 0.909i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.08 - 0.909i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.459205053943255757750043562125, −8.617073162850434549119662257261, −8.025293107842212973742196627214, −7.09070640146375918803291402947, −5.60040353094787667422337801153, −4.81958902432599344177310013349, −4.22872587354002815513515369891, −3.46789926067049238530870608720, −2.56457240142610405216011175310, −1.74604162140077942171069522001,
2.07527952267958783887988050643, 3.03669229030833191318267369680, 3.60026943015652998909238384991, 4.70099106110708516746914141548, 5.60350246482984774195383997464, 6.65683833250417772839819760457, 7.16364985983747647654897661927, 7.75077807100674449674393605284, 8.382511688466343796965118372962, 9.564011820627474307912052632930