L(s) = 1 | + (0.309 − 0.951i)3-s + 1.61·7-s + (−0.951 + 1.30i)13-s + (0.587 − 0.190i)19-s + (0.500 − 1.53i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.190 − 0.587i)29-s + (0.951 − 0.309i)31-s + (0.587 − 0.809i)37-s + (0.951 + 1.30i)39-s + 0.618·43-s + (−0.5 + 1.53i)47-s + 1.61·49-s + (−0.951 − 0.309i)53-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)3-s + 1.61·7-s + (−0.951 + 1.30i)13-s + (0.587 − 0.190i)19-s + (0.500 − 1.53i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.190 − 0.587i)29-s + (0.951 − 0.309i)31-s + (0.587 − 0.809i)37-s + (0.951 + 1.30i)39-s + 0.618·43-s + (−0.5 + 1.53i)47-s + 1.61·49-s + (−0.951 − 0.309i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.783315836\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783315836\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.221307232432302404824957489271, −7.80915052021029706615614760468, −7.32618924120938795641078568337, −6.52503560481823514820035519029, −5.61881503924697154934373624566, −4.58880446137308658945649355976, −4.31360695967527327994523736947, −2.71172869604517473426881947558, −1.97322287210696176740920862652, −1.29783510108590529105028596367,
1.21353372159341419248320091114, 2.43685760503564186696678873216, 3.31378896328819707873125047547, 4.27926016418992816742135405224, 4.94359282659929912133959401486, 5.32060289370328324404104145303, 6.48851456591399040887447506492, 7.49299795971589265371172666704, 8.115337255113439146147754447597, 8.513923029490471018549814470936