L(s) = 1 | + 2.66·2-s + 3-s + 5.10·4-s + (−1.18 + 1.89i)5-s + 2.66·6-s + (−2.09 + 1.61i)7-s + 8.26·8-s + 9-s + (−3.15 + 5.05i)10-s + (1.90 + 2.71i)11-s + 5.10·12-s − 0.864i·13-s + (−5.59 + 4.29i)14-s + (−1.18 + 1.89i)15-s + 11.8·16-s − 1.53i·17-s + ⋯ |
L(s) = 1 | + 1.88·2-s + 0.577·3-s + 2.55·4-s + (−0.530 + 0.847i)5-s + 1.08·6-s + (−0.793 + 0.608i)7-s + 2.92·8-s + 0.333·9-s + (−0.998 + 1.59i)10-s + (0.573 + 0.819i)11-s + 1.47·12-s − 0.239i·13-s + (−1.49 + 1.14i)14-s + (−0.306 + 0.489i)15-s + 2.95·16-s − 0.372i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.501539842\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.501539842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (1.18 - 1.89i)T \) |
| 7 | \( 1 + (2.09 - 1.61i)T \) |
| 11 | \( 1 + (-1.90 - 2.71i)T \) |
good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 13 | \( 1 + 0.864iT - 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 23 | \( 1 + 7.32iT - 23T^{2} \) |
| 29 | \( 1 - 1.42iT - 29T^{2} \) |
| 31 | \( 1 - 0.551iT - 31T^{2} \) |
| 37 | \( 1 + 5.97iT - 37T^{2} \) |
| 41 | \( 1 - 0.607T + 41T^{2} \) |
| 43 | \( 1 - 0.823T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 - 8.21iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 - 7.78iT - 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 3.89iT - 73T^{2} \) |
| 79 | \( 1 - 8.94iT - 79T^{2} \) |
| 83 | \( 1 + 13.1iT - 83T^{2} \) |
| 89 | \( 1 - 7.51iT - 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−10.21263510265435140343841929354, −9.080973330256305994285565494186, −7.889126624201811382654657137186, −6.89386408400207160641094074523, −6.60954553630561961103673819140, −5.59556516626588670982103245720, −4.40481453438588497821572191385, −3.80960832735681303924683551265, −2.81220282165116566821209242464, −2.27013476502672255673435288591,
1.41673524535603789678505679492, 2.89819331663975789010448789584, 3.86548683638107388909770826405, 4.09438542505300180340876563895, 5.30775518253480708427686282860, 6.15214862900974404554905341766, 6.98651865720796801362458419252, 7.78617893400367211663962876480, 8.778247516995940695704694610419, 9.787607461313831267942827354436