L(s) = 1 | + (−0.707 − 1.22i)2-s + 3-s + (−0.997 + 1.73i)4-s − 3.14i·5-s + (−0.707 − 1.22i)6-s + 1.38·7-s + (2.82 − 0.00564i)8-s + 9-s + (−3.85 + 2.22i)10-s + 2.19·11-s + (−0.997 + 1.73i)12-s − 1.11i·13-s + (−0.978 − 1.69i)14-s − 3.14i·15-s + (−2.00 − 3.45i)16-s + 0.639·17-s + ⋯ |
L(s) = 1 | + (−0.500 − 0.865i)2-s + 0.577·3-s + (−0.498 + 0.866i)4-s − 1.40i·5-s + (−0.289 − 0.499i)6-s + 0.522·7-s + (0.999 − 0.00199i)8-s + 0.333·9-s + (−1.21 + 0.704i)10-s + 0.660·11-s + (−0.288 + 0.500i)12-s − 0.308i·13-s + (−0.261 − 0.452i)14-s − 0.812i·15-s + (−0.502 − 0.864i)16-s + 0.155·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.764788 - 1.29890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.764788 - 1.29890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (-8.18 + 0.128i)T \) |
good | 5 | \( 1 + 3.14iT - 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + 1.11iT - 13T^{2} \) |
| 17 | \( 1 - 0.639T + 17T^{2} \) |
| 19 | \( 1 - 2.14iT - 19T^{2} \) |
| 23 | \( 1 + 5.50iT - 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.53iT - 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 - 2.88iT - 47T^{2} \) |
| 53 | \( 1 + 0.00678iT - 53T^{2} \) |
| 59 | \( 1 + 0.119iT - 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 71 | \( 1 + 5.27iT - 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 6.31iT - 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 5.12iT - 97T^{2} \) |
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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
−9.900799243292082530495253797524, −9.055908279225297625066136022313, −8.388495351878581708549300265435, −8.045624042136276331786689605454, −6.68295703132267265464463275471, −5.06372459279684149933930120193, −4.43663724485986579459092392867, −3.35776677039187325540077317108, −1.95104849666617265850211357302, −0.934171184816826964522621978946,
1.62210343213331274614363355292, 3.03849383533536302782331287994, 4.22809285100637547895596279298, 5.40407023046862312169835295360, 6.70708420730731294247784984613, 6.91466870653326835953646528623, 7.974901750470324641475550955123, 8.674844087018656340544963234210, 9.665849206991245948083493681129, 10.27886230197116863000786702054