L(s) = 1 | + 0.254·2-s − i·3-s − 1.93·4-s − 0.458·5-s − 0.254i·6-s + (1.07 + 2.41i)7-s − 8-s − 9-s − 0.116·10-s + 1.34i·11-s + 1.93i·12-s + 4.42i·13-s + (0.272 + 0.614i)14-s + 0.458i·15-s + 3.61·16-s + 7.09·17-s + ⋯ |
L(s) = 1 | + 0.179·2-s − 0.577i·3-s − 0.967·4-s − 0.204·5-s − 0.103i·6-s + (0.405 + 0.914i)7-s − 0.353·8-s − 0.333·9-s − 0.0368·10-s + 0.405i·11-s + 0.558i·12-s + 1.22i·13-s + (0.0728 + 0.164i)14-s + 0.118i·15-s + 0.904·16-s + 1.72·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04673 + 0.444512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04673 + 0.444512i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.07 - 2.41i)T \) |
| 23 | \( 1 + (4.44 - 1.80i)T \) |
good | 2 | \( 1 - 0.254T + 2T^{2} \) |
| 5 | \( 1 + 0.458T + 5T^{2} \) |
| 11 | \( 1 - 1.34iT - 11T^{2} \) |
| 13 | \( 1 - 4.42iT - 13T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 + 4.31iT - 31T^{2} \) |
| 37 | \( 1 - 8.56iT - 37T^{2} \) |
| 41 | \( 1 - 3.81iT - 41T^{2} \) |
| 43 | \( 1 - 7.18iT - 43T^{2} \) |
| 47 | \( 1 - 7.23iT - 47T^{2} \) |
| 53 | \( 1 + 8.35iT - 53T^{2} \) |
| 59 | \( 1 + 8.10iT - 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 5.84iT - 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 7.04iT - 73T^{2} \) |
| 79 | \( 1 - 6.86iT - 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−11.58275220935982815930709406185, −9.793512515238799584277929428817, −9.478485507046573362245390835073, −8.175314363533576140029181655507, −7.78817623485236255135822487363, −6.30027767146910708281340740530, −5.42615975649888714392222794534, −4.45224955099655560302130781008, −3.16135255178487280139185860636, −1.53335535957880372390536042589,
0.75318145690819381383556658364, 3.31076798953150668395354237144, 3.95564213444854705389079278858, 5.16541029374063012178142135263, 5.76922266727565427745632788641, 7.58093660426084873950626414041, 8.076028601693881119378159269884, 9.146891819805436966522595405368, 10.28344925911979451542774731037, 10.40041160413968983857020854852