L(s) = 1 | + i·3-s + (−0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (0.587 − 0.809i)7-s − 9-s + (0.309 − 0.951i)11-s + (0.587 − 0.809i)12-s + (1.53 + 0.5i)13-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.587 + 1.80i)17-s + (0.951 + 0.309i)20-s + (0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s − i·27-s + (−0.951 + 0.309i)28-s + ⋯ |
L(s) = 1 | + i·3-s + (−0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (0.587 − 0.809i)7-s − 9-s + (0.309 − 0.951i)11-s + (0.587 − 0.809i)12-s + (1.53 + 0.5i)13-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.587 + 1.80i)17-s + (0.951 + 0.309i)20-s + (0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s − i·27-s + (−0.951 + 0.309i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8241884499\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8241884499\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.30629583605147925640408360911, −9.181370971711609494343506437538, −8.360327899012041873750345048808, −8.134204286718524604061308731691, −6.49912601546564038660490967438, −5.79170756857219151004450822739, −4.67566409800490754307966441796, −3.81278511791054016301068706905, −3.60757696700878178513388455860, −1.17991437844168444457522533349,
1.03573948806927895095104719075, 2.68254238231659724828567889576, 3.69891385183011610472171986263, 4.81940596117408294880341602861, 5.53175803400055859364427336533, 6.83324854739648502792884941192, 7.66080590514410180580751706555, 8.217843620370051394417854895172, 8.819730704506362399206017274541, 9.571036566776949598036118128697