L(s) = 1 | + (−0.114 − 0.0855i)3-s + (−0.909 + 0.415i)4-s + (−0.755 + 0.654i)5-s + (−0.275 − 0.939i)9-s + (−0.540 − 0.841i)11-s + (0.139 + 0.0303i)12-s + (0.142 − 0.0101i)15-s + (0.654 − 0.755i)16-s + (0.415 − 0.909i)20-s + (0.841 − 0.540i)23-s + (0.142 − 0.989i)25-s + (−0.0987 + 0.264i)27-s + (0.0801 − 0.557i)31-s + (−0.0101 + 0.142i)33-s + (0.641 + 0.740i)36-s + (0.898 − 1.64i)37-s + ⋯ |
L(s) = 1 | + (−0.114 − 0.0855i)3-s + (−0.909 + 0.415i)4-s + (−0.755 + 0.654i)5-s + (−0.275 − 0.939i)9-s + (−0.540 − 0.841i)11-s + (0.139 + 0.0303i)12-s + (0.142 − 0.0101i)15-s + (0.654 − 0.755i)16-s + (0.415 − 0.909i)20-s + (0.841 − 0.540i)23-s + (0.142 − 0.989i)25-s + (−0.0987 + 0.264i)27-s + (0.0801 − 0.557i)31-s + (−0.0101 + 0.142i)33-s + (0.641 + 0.740i)36-s + (0.898 − 1.64i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4926582363\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4926582363\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.755 - 0.654i)T \) |
| 11 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (-0.841 + 0.540i)T \) |
good | 2 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 3 | \( 1 + (0.114 + 0.0855i)T + (0.281 + 0.959i)T^{2} \) |
| 7 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 13 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 17 | \( 1 + (0.755 - 0.654i)T^{2} \) |
| 19 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.0801 + 0.557i)T + (-0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (-0.898 + 1.64i)T + (-0.540 - 0.841i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 47 | \( 1 + (1.24 + 1.24i)T + iT^{2} \) |
| 53 | \( 1 + (-0.697 + 0.0498i)T + (0.989 - 0.142i)T^{2} \) |
| 59 | \( 1 + (1.37 - 1.19i)T + (0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (-1.94 + 0.424i)T + (0.909 - 0.415i)T^{2} \) |
| 71 | \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (-0.755 - 0.654i)T^{2} \) |
| 79 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 89 | \( 1 + (0.215 + 1.49i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 - 0.459i)T + (0.540 - 0.841i)T^{2} \) |
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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
−9.575014579219095114723421146297, −8.809682604487343869099502970311, −8.138319547427436238105054398152, −7.37107896896091002019015631645, −6.42063493646047922711969235514, −5.50183333163984470903916368031, −4.38992730005716411615374072053, −3.53870287791063571191273317644, −2.83951579124534615099245020820, −0.47288489317332877783183378660,
1.44386304369985993706027701968, 3.07083287344622959724550832747, 4.38258111430182724145904510961, 4.87890061456298884698386254265, 5.51539838499732834679518109833, 6.84403081458257562077730704041, 7.997818204272257664867789423794, 8.230887673709361572730830726409, 9.344018125845290092059404795699, 9.893170520138620857388475715201