L(s) = 1 | + (−0.978 + 0.207i)3-s + (−0.104 + 0.994i)4-s + 5-s + (0.913 − 0.406i)9-s + (0.669 + 0.743i)11-s + (−0.104 − 0.994i)12-s + (−0.978 + 0.207i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)20-s + (1.58 − 0.336i)23-s + 25-s + (−0.809 + 0.587i)27-s + (1.22 + 0.544i)31-s + (−0.809 − 0.587i)33-s + (0.309 + 0.951i)36-s + (−0.309 − 0.951i)37-s + ⋯ |
L(s) = 1 | + (−0.978 + 0.207i)3-s + (−0.104 + 0.994i)4-s + 5-s + (0.913 − 0.406i)9-s + (0.669 + 0.743i)11-s + (−0.104 − 0.994i)12-s + (−0.978 + 0.207i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)20-s + (1.58 − 0.336i)23-s + 25-s + (−0.809 + 0.587i)27-s + (1.22 + 0.544i)31-s + (−0.809 − 0.587i)33-s + (0.309 + 0.951i)36-s + (−0.309 − 0.951i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.096682686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.096682686\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + (-0.669 - 0.743i)T \) |
good | 2 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \) |
| 29 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 0.544i)T + (0.669 + 0.743i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 53 | \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (1.30 - 1.45i)T + (-0.104 - 0.994i)T^{2} \) |
| 61 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 67 | \( 1 + (0.190 + 0.0850i)T + (0.669 + 0.743i)T^{2} \) |
| 71 | \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 83 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 89 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-1.22 + 0.544i)T + (0.669 - 0.743i)T^{2} \) |
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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
−9.320045632180466271794135256883, −8.681417830652494103742712136169, −7.56313566188084008622679144304, −6.72612804276870582011530672927, −6.42045663193914660849473363845, −5.15564184878782810373884244864, −4.70947308103997580972938868409, −3.69959529415523795914345385408, −2.61537426290941559899913345331, −1.36851558810799209782976405018,
0.988861015885058307228252231399, 1.74890988081040408612731150279, 3.11693920674292507162562823943, 4.67038331740424333419290591717, 5.05799800669720770065754605978, 6.10367518175023266605675135746, 6.27926744969575668108267556061, 7.06908796902213714945377607382, 8.290984839840925573675480257987, 9.314740180116530819737189763647