L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.63 + 0.189i)7-s + 0.999·8-s + (−2.55 − 1.47i)11-s + 3.93·13-s + (1.48 + 2.19i)14-s + (−0.5 − 0.866i)16-s + (0.346 + 0.199i)17-s + (−0.0305 + 0.0176i)19-s + 2.94i·22-s + (1.86 + 3.23i)23-s + (−1.96 − 3.40i)26-s + (1.15 − 2.38i)28-s − 8.89i·29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.997 + 0.0716i)7-s + 0.353·8-s + (−0.769 − 0.444i)11-s + 1.09·13-s + (0.396 + 0.585i)14-s + (−0.125 − 0.216i)16-s + (0.0839 + 0.0484i)17-s + (−0.00700 + 0.00404i)19-s + 0.628i·22-s + (0.389 + 0.673i)23-s + (−0.385 − 0.667i)26-s + (0.218 − 0.449i)28-s − 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6117924853\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6117924853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.189i)T \) |
good | 11 | \( 1 + (2.55 + 1.47i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + (-0.346 - 0.199i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0305 - 0.0176i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-0.717 - 0.414i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.86 - 3.96i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.03iT - 43T^{2} \) |
| 47 | \( 1 + (5.02 - 2.90i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.14 + 3.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.78 + 4.82i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.97 - 5.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 6.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.93iT - 71T^{2} \) |
| 73 | \( 1 + (0.171 - 0.297i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.748679925157395628478921680767, −8.340241329063784385964062386038, −7.45686985200903527028662630966, −6.53809119226391635500962518171, −5.86757140241846843003884005583, −4.95713796967914377703176437697, −3.76907726440024110242718147394, −3.25796915075329272736895513582, −2.31891230570068307804751129728, −1.03236809458050961050292931789,
0.24861432756776822433932023453, 1.66571667756371373200642055637, 2.98548552421179873907049105026, 3.76237861132350445412757294782, 4.89736911671419365075431986080, 5.57116787512432617904017166032, 6.48604830225583808833657463644, 6.94097692720658197964966311160, 7.74249750761879843241449312642, 8.667485936567323508775938896138