L(s) = 1 | + 1.76i·3-s + (−0.432 − 2.19i)5-s + i·7-s − 0.103·9-s − 0.626·11-s − 5.49i·13-s + (3.86 − 0.761i)15-s − 0.896i·17-s − 6.38·19-s − 1.76·21-s + 3.72i·23-s + (−4.62 + 1.89i)25-s + 5.10i·27-s − 7.87·29-s − 7.52·31-s + ⋯ |
L(s) = 1 | + 1.01i·3-s + (−0.193 − 0.981i)5-s + 0.377i·7-s − 0.0343·9-s − 0.188·11-s − 1.52i·13-s + (0.997 − 0.196i)15-s − 0.217i·17-s − 1.46·19-s − 0.384·21-s + 0.777i·23-s + (−0.925 + 0.379i)25-s + 0.982i·27-s − 1.46·29-s − 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1500808559\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1500808559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.432 + 2.19i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 1.76iT - 3T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 + 5.49iT - 13T^{2} \) |
| 17 | \( 1 + 0.896iT - 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 - 1.72iT - 43T^{2} \) |
| 47 | \( 1 - 5.87iT - 47T^{2} \) |
| 53 | \( 1 - 6.77iT - 53T^{2} \) |
| 59 | \( 1 - 0.593T + 59T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 3.72iT - 73T^{2} \) |
| 79 | \( 1 + 5.67T + 79T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
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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
−9.314346318730559705746141808353, −8.911749915440638161103766385270, −7.974102074984987695442085811393, −7.39505508519617777270322412281, −5.91388098460740276286844239841, −5.44881986698660598646420491635, −4.60338808550463143915649929152, −3.92344312383140211875787605508, −2.96360464399758446891355516409, −1.56241598410035600031257836793,
0.04863971539076221601237046342, 1.81757199203017704418357965273, 2.32180871166509327186218302315, 3.79320915617729091840821074728, 4.31398863400467356048810683756, 5.78810694418831548093745231933, 6.54684812616432003709421648049, 7.07219151205645246256062822470, 7.56598148967142036134531520700, 8.499645473546821915281645780434