L(s) = 1 | − 1.41·2-s + (1 − 1.41i)3-s + i·5-s + (−1.41 + 2.00i)6-s + i·7-s + 2.82·8-s + (−1.00 − 2.82i)9-s − 1.41i·10-s + (−3 − 1.41i)11-s + 6.24i·13-s − 1.41i·14-s + (1.41 + i)15-s − 4.00·16-s − 0.171·17-s + (1.41 + 4.00i)18-s − 5.24i·19-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (0.577 − 0.816i)3-s + 0.447i·5-s + (−0.577 + 0.816i)6-s + 0.377i·7-s + 0.999·8-s + (−0.333 − 0.942i)9-s − 0.447i·10-s + (−0.904 − 0.426i)11-s + 1.73i·13-s − 0.377i·14-s + (0.365 + 0.258i)15-s − 1.00·16-s − 0.0416·17-s + (0.333 + 0.942i)18-s − 1.20i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4527423979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4527423979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 13 | \( 1 - 6.24iT - 13T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 + 5.24iT - 19T^{2} \) |
| 23 | \( 1 + 7.24iT - 23T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 + 0.242T + 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 5.24iT - 43T^{2} \) |
| 47 | \( 1 + 13.0iT - 47T^{2} \) |
| 53 | \( 1 - 1.58iT - 53T^{2} \) |
| 59 | \( 1 + 6.89iT - 59T^{2} \) |
| 61 | \( 1 + 0.757iT - 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 2.24iT - 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.183000035098062507217929303383, −8.640896599696818699266394564433, −8.078927569222376065701099474376, −6.91203704590542942370675355705, −6.73041019965430853882589890832, −5.22339212245830237446049623334, −4.06983312236944761954448656367, −2.67971866236220210564290020060, −1.86924436097575112259862729840, −0.26149358816505158561078031671,
1.45682568409534002257902270674, 2.95791552281750598186005063717, 3.99769320202935130891428941274, 5.00448469238100960787864606433, 5.65339590942926893933508536536, 7.48531947704859778106705494876, 7.915650443076854751973526706339, 8.442544382379299109342295347445, 9.481153703904381174760094350748, 9.995259322398124904836472461543