L(s) = 1 | − i·3-s − i·5-s − 1.53·7-s − 9-s + 3.80·11-s + 2.39·13-s − 15-s − 2.51i·17-s − 3.13·19-s + 1.53i·21-s + (−3.77 + 2.95i)23-s − 25-s + i·27-s − 1.76·29-s + 3.33i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.579·7-s − 0.333·9-s + 1.14·11-s + 0.663·13-s − 0.258·15-s − 0.610i·17-s − 0.719·19-s + 0.334i·21-s + (−0.787 + 0.616i)23-s − 0.200·25-s + 0.192i·27-s − 0.328·29-s + 0.599i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4576446804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4576446804\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (3.77 - 2.95i)T \) |
good | 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 2.51iT - 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 - 3.33iT - 31T^{2} \) |
| 37 | \( 1 + 3.84iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 0.753T + 43T^{2} \) |
| 47 | \( 1 + 9.86iT - 47T^{2} \) |
| 53 | \( 1 - 5.02iT - 53T^{2} \) |
| 59 | \( 1 - 11.1iT - 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 + 1.33T + 67T^{2} \) |
| 71 | \( 1 + 7.54iT - 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 - 2.23T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 + 1.26iT - 97T^{2} \) |
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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
−7.78807214039710078317304532490, −6.84530549737978834170406793282, −6.49065162115392003582476829603, −5.71662355699637806198620204822, −4.89939885576858291923006314870, −3.86157490326189496515759481512, −3.36449872904604690494273936825, −2.08904109418698117249590188340, −1.32507404910949881016292905094, −0.11839501194736505142265799426,
1.45224496784654582478276248846, 2.51502031584048633036552044867, 3.60651871153936367639074783053, 3.88530431542795455076431532995, 4.81218556847413078801324826836, 5.87379757622366134984264621826, 6.44931019314806707049875870704, 6.81982798307638692630139953209, 8.117384340970694513497151884748, 8.441808192663913521390055721639