L(s) = 1 | + 3-s + (−0.220 + 0.220i)5-s + (0.834 + 0.834i)7-s + 9-s + (−4.19 + 4.19i)11-s + (1.68 − 3.18i)13-s + (−0.220 + 0.220i)15-s + 5.40i·17-s + (−1.07 − 1.07i)19-s + (0.834 + 0.834i)21-s + 8.62·23-s + 4.90i·25-s + 27-s + 7.07i·29-s + (−0.834 + 0.834i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.0986 + 0.0986i)5-s + (0.315 + 0.315i)7-s + 0.333·9-s + (−1.26 + 1.26i)11-s + (0.466 − 0.884i)13-s + (−0.0569 + 0.0569i)15-s + 1.31i·17-s + (−0.247 − 0.247i)19-s + (0.182 + 0.182i)21-s + 1.79·23-s + 0.980i·25-s + 0.192·27-s + 1.31i·29-s + (−0.149 + 0.149i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.828384190\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.828384190\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (-1.68 + 3.18i)T \) |
good | 5 | \( 1 + (0.220 - 0.220i)T - 5iT^{2} \) |
| 7 | \( 1 + (-0.834 - 0.834i)T + 7iT^{2} \) |
| 11 | \( 1 + (4.19 - 4.19i)T - 11iT^{2} \) |
| 17 | \( 1 - 5.40iT - 17T^{2} \) |
| 19 | \( 1 + (1.07 + 1.07i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.62T + 23T^{2} \) |
| 29 | \( 1 - 7.07iT - 29T^{2} \) |
| 31 | \( 1 + (0.834 - 0.834i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.93 + 3.93i)T + 37iT^{2} \) |
| 41 | \( 1 + (-7.68 - 7.68i)T + 41iT^{2} \) |
| 43 | \( 1 - 4.27iT - 43T^{2} \) |
| 47 | \( 1 + (6.33 + 6.33i)T + 47iT^{2} \) |
| 53 | \( 1 - 0.517iT - 53T^{2} \) |
| 59 | \( 1 + (-3.64 + 3.64i)T - 59iT^{2} \) |
| 61 | \( 1 - 8.80iT - 61T^{2} \) |
| 67 | \( 1 + (3.20 + 3.20i)T + 67iT^{2} \) |
| 71 | \( 1 + (-9.26 + 9.26i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.70 - 1.70i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.7iT - 79T^{2} \) |
| 83 | \( 1 + (-5.49 - 5.49i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.33 - 2.33i)T - 89iT^{2} \) |
| 97 | \( 1 + (1.70 + 1.70i)T + 97iT^{2} \) |
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
−9.850408830377102976245601120250, −8.929476467468167295372618841657, −8.231179331890855756667448476214, −7.51552644845915338364898510056, −6.75391611914721524590947969838, −5.42991316279560331087793092478, −4.85960918655913196730369890617, −3.58294644814444247284268384908, −2.68452208127937582555327115158, −1.55069558569409292655550127943,
0.74530898034957490469531215831, 2.37474937385416891496449025554, 3.23285234969982062877680718089, 4.35989399143901582774828687539, 5.20470886696753992480287774988, 6.26129816067339196269947304561, 7.25919103134475563446822476836, 7.983075088711561264134914696814, 8.713754905722510957675762492488, 9.368439804722678523156205312654