L(s) = 1 | + (−0.379 + 1.36i)2-s + (−1.71 − 1.03i)4-s + (0.463 − 0.463i)5-s − 1.85·7-s + (2.05 − 1.93i)8-s + (0.455 + 0.808i)10-s + (3.73 + 3.73i)11-s + (−4.31 + 4.31i)13-s + (0.705 − 2.53i)14-s + (1.85 + 3.54i)16-s + 6.55i·17-s + (−1.31 − 1.31i)19-s + (−1.27 + 0.314i)20-s + (−6.50 + 3.67i)22-s − 0.727i·23-s + ⋯ |
L(s) = 1 | + (−0.268 + 0.963i)2-s + (−0.855 − 0.517i)4-s + (0.207 − 0.207i)5-s − 0.702·7-s + (0.727 − 0.685i)8-s + (0.144 + 0.255i)10-s + (1.12 + 1.12i)11-s + (−1.19 + 1.19i)13-s + (0.188 − 0.676i)14-s + (0.464 + 0.885i)16-s + 1.59i·17-s + (−0.301 − 0.301i)19-s + (−0.284 + 0.0702i)20-s + (−1.38 + 0.782i)22-s − 0.151i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.299475 + 0.827515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.299475 + 0.827515i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.379 - 1.36i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.463 + 0.463i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 + (-3.73 - 3.73i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.31 - 4.31i)T - 13iT^{2} \) |
| 17 | \( 1 - 6.55iT - 17T^{2} \) |
| 19 | \( 1 + (1.31 + 1.31i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.727iT - 23T^{2} \) |
| 29 | \( 1 + (-0.896 - 0.896i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.25iT - 31T^{2} \) |
| 37 | \( 1 + (-3.98 - 3.98i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.10T + 41T^{2} \) |
| 43 | \( 1 + (-1.09 + 1.09i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.81T + 47T^{2} \) |
| 53 | \( 1 + (0.712 - 0.712i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.86 - 1.86i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.98 + 5.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (8.48 + 8.48i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.72iT - 73T^{2} \) |
| 79 | \( 1 + 0.199iT - 79T^{2} \) |
| 83 | \( 1 + (-9.97 + 9.97i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 - 9.81T + 97T^{2} \) |
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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
−11.54505666468762845471413682600, −10.13467820793753723248042234183, −9.526867237578956595700851246195, −8.934936908299404656816458352936, −7.67167330581304755409745393777, −6.76078883327773408948847737709, −6.18763431172522211515743824662, −4.77254640015118083865063935170, −3.98244937696698821588396137198, −1.79067411149677252131777541973,
0.62741725043958894552320479217, 2.61976480433013513678646886738, 3.41068856484324319667563574620, 4.77918195698933993964317162837, 5.98753810750850392625697556948, 7.21305460647422486000259058782, 8.292466694243986208884988834368, 9.310011188534386387421471469603, 9.878553930672474271088152118129, 10.76984465005950361987324147037