L(s) = 1 | + (0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (0.690 + 0.951i)7-s + (−0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s + (0.309 − 0.951i)16-s + (−1.80 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)20-s + 1.90i·23-s + (−0.809 + 0.587i)25-s + (1.11 + 0.363i)28-s + (−0.690 + 0.951i)35-s + (−0.809 − 0.587i)36-s − 1.90i·43-s + 44-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (0.690 + 0.951i)7-s + (−0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s + (0.309 − 0.951i)16-s + (−1.80 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)20-s + 1.90i·23-s + (−0.809 + 0.587i)25-s + (1.11 + 0.363i)28-s + (−0.690 + 0.951i)35-s + (−0.809 − 0.587i)36-s − 1.90i·43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.328262453\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328262453\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - 1.90iT - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 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 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
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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
−10.21113054886577532933995941339, −9.246522579748464136010862408122, −8.825231824530437302729473624046, −7.25295890388695595125625171485, −6.81245314741722414884067953495, −6.02246635816760699225180128623, −5.21953703049033772856713006860, −3.83684259835961469185248358774, −2.52399645412985078657528973218, −1.87194181839078801009391574155,
1.53864732888383593462439974632, 2.52164043880825133351640873521, 4.22382880013617333151397361241, 4.50273962629193106970797731922, 6.03483358937890099689493189535, 6.62799677270334789006967904469, 7.81440300553720821496164026952, 8.364660757354038905880500818732, 8.945204764197273591001717889676, 10.38325839956642161863305254780