L(s) = 1 | + (−0.260 + 0.260i)2-s + (−0.826 − 1.52i)3-s + 1.86i·4-s + (0.611 + 0.180i)6-s + (−0.707 − 0.707i)7-s + (−1.00 − 1.00i)8-s + (−1.63 + 2.51i)9-s − 3.38i·11-s + (2.83 − 1.54i)12-s + (−1.59 + 1.59i)13-s + 0.368·14-s − 3.20·16-s + (−0.140 + 0.140i)17-s + (−0.230 − 1.07i)18-s − 7.34i·19-s + ⋯ |
L(s) = 1 | + (−0.184 + 0.184i)2-s + (−0.477 − 0.878i)3-s + 0.932i·4-s + (0.249 + 0.0738i)6-s + (−0.267 − 0.267i)7-s + (−0.355 − 0.355i)8-s + (−0.544 + 0.838i)9-s − 1.02i·11-s + (0.819 − 0.445i)12-s + (−0.442 + 0.442i)13-s + 0.0983·14-s − 0.801·16-s + (−0.0341 + 0.0341i)17-s + (−0.0542 − 0.254i)18-s − 1.68i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.144657 - 0.378828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.144657 - 0.378828i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.826 + 1.52i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (0.260 - 0.260i)T - 2iT^{2} \) |
| 11 | \( 1 + 3.38iT - 11T^{2} \) |
| 13 | \( 1 + (1.59 - 1.59i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.140 - 0.140i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.34iT - 19T^{2} \) |
| 23 | \( 1 + (2.21 + 2.21i)T + 23iT^{2} \) |
| 29 | \( 1 + 9.49T + 29T^{2} \) |
| 31 | \( 1 - 0.922T + 31T^{2} \) |
| 37 | \( 1 + (5.91 + 5.91i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.39iT - 41T^{2} \) |
| 43 | \( 1 + (0.864 - 0.864i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.651 - 0.651i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.54 + 6.54i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.25T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 + (-0.815 - 0.815i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.77iT - 71T^{2} \) |
| 73 | \( 1 + (-4.80 + 4.80i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.41iT - 79T^{2} \) |
| 83 | \( 1 + (-6.26 - 6.26i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + (-6.71 - 6.71i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.87106973016819798044159413609, −9.373324438508521638874877942886, −8.608085724301474203601418518561, −7.66563407232733989010420177698, −6.97372078491655678893032713830, −6.19200375930075878637188255902, −4.92552606883980296074893552898, −3.54406884077322847349767370885, −2.32837060391583050993418102030, −0.25339454953567200242648595617,
1.85136750564701038751420504675, 3.50356489099745560070458077581, 4.74457092784149894741012380469, 5.57364272092875417625789733436, 6.31189394448554660798567686282, 7.61593865050526716991310656127, 8.927340702650658543023679985057, 9.797614878456276693142400712621, 10.09711375639760593771939074642, 10.98034814328837303738610040018