L(s) = 1 | − 2-s + i·3-s − 4-s − 3.33·5-s − i·6-s + (−2.60 − 0.468i)7-s + 3·8-s − 9-s + 3.33·10-s + 4.27i·11-s − i·12-s − 3.12i·13-s + (2.60 + 0.468i)14-s − 3.33i·15-s − 16-s + 3.33·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s − 0.5·4-s − 1.49·5-s − 0.408i·6-s + (−0.984 − 0.176i)7-s + 1.06·8-s − 0.333·9-s + 1.05·10-s + 1.28i·11-s − 0.288i·12-s − 0.866i·13-s + (0.695 + 0.125i)14-s − 0.861i·15-s − 0.250·16-s + 0.808·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.384758 - 0.100205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.384758 - 0.100205i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (2.60 + 0.468i)T \) |
| 23 | \( 1 + (-1.56 + 4.53i)T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 11 | \( 1 - 4.27iT - 11T^{2} \) |
| 13 | \( 1 + 3.12iT - 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 - 11.1iT - 41T^{2} \) |
| 43 | \( 1 - 0.936iT - 43T^{2} \) |
| 47 | \( 1 - 7.12iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 7.12iT - 59T^{2} \) |
| 61 | \( 1 - 1.87T + 61T^{2} \) |
| 67 | \( 1 + 4.68iT - 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 6.24iT - 73T^{2} \) |
| 79 | \( 1 + 16.1iT - 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 1.87T + 97T^{2} \) |
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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
−10.53777385258310626164397047776, −10.01195315863652688468613126375, −9.252523489762105485068472057684, −8.086437202301223030877082324703, −7.71807472023291929318848298260, −6.48875594044530363186917951922, −4.81661573548158567878621721260, −4.17963334364974551901377929254, −3.08026622977014306256533459985, −0.45753798898387082510068007954,
0.898173715919261999745550697673, 3.18171637714577314613753289873, 4.03998191228707960984719580580, 5.47627767740253056541918452392, 6.82075294967096345594439095033, 7.51077201961611092230534124617, 8.636296696118984061453038798811, 8.796153777744982815517755256595, 10.14982225082689819017951082743, 11.01433707098437052843536167616