| L(s) = 1 | + (0.813 + 1.15i)2-s + (0.130 − 0.991i)3-s + (−0.676 + 1.88i)4-s + (0.0451 + 0.343i)5-s + (1.25 − 0.655i)6-s + (−1.63 + 2.08i)7-s + (−2.72 + 0.748i)8-s + (−0.965 − 0.258i)9-s + (−0.360 + 0.331i)10-s + (1.22 + 1.60i)11-s + (1.77 + 0.916i)12-s + (−1.30 + 0.539i)13-s + (−3.73 − 0.196i)14-s + 0.346·15-s + (−3.08 − 2.54i)16-s + (−2.39 + 4.14i)17-s + ⋯ |
| L(s) = 1 | + (0.575 + 0.817i)2-s + (0.0753 − 0.572i)3-s + (−0.338 + 0.941i)4-s + (0.0202 + 0.153i)5-s + (0.511 − 0.267i)6-s + (−0.617 + 0.786i)7-s + (−0.964 + 0.264i)8-s + (−0.321 − 0.0862i)9-s + (−0.113 + 0.104i)10-s + (0.370 + 0.482i)11-s + (0.513 + 0.264i)12-s + (−0.361 + 0.149i)13-s + (−0.998 − 0.0526i)14-s + 0.0893·15-s + (−0.771 − 0.636i)16-s + (−0.580 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375657 + 1.33038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375657 + 1.33038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.813 - 1.15i)T \) |
| 3 | \( 1 + (-0.130 + 0.991i)T \) |
| 7 | \( 1 + (1.63 - 2.08i)T \) |
| good | 5 | \( 1 + (-0.0451 - 0.343i)T + (-4.82 + 1.29i)T^{2} \) |
| 11 | \( 1 + (-1.22 - 1.60i)T + (-2.84 + 10.6i)T^{2} \) |
| 13 | \( 1 + (1.30 - 0.539i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (2.39 - 4.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.89 - 2.46i)T + (-4.91 - 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.523 + 1.95i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.89 - 0.786i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (3.64 - 6.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.01 + 0.134i)T + (35.7 - 9.57i)T^{2} \) |
| 41 | \( 1 + (-6.88 - 6.88i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.814 + 1.96i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (6.59 - 3.80i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.77 + 5.19i)T + (13.7 - 51.1i)T^{2} \) |
| 59 | \( 1 + (0.739 + 0.963i)T + (-15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-3.06 + 3.98i)T + (-15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (-10.5 - 1.38i)T + (64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (9.42 - 9.42i)T - 71iT^{2} \) |
| 73 | \( 1 + (-8.29 + 2.22i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.64 + 9.78i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.53 - 15.7i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.0955 - 0.0256i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 18.2iT - 97T^{2} \) |
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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
−11.04104174370822305958784958823, −9.762224669769022735845546713140, −8.825323959923043205624942714327, −8.227197634756203337684561786200, −7.03122945135183085152978455452, −6.51008277503681571364951461479, −5.68258852598593528477253204363, −4.53787185758608838056948921574, −3.35636387110564919357259511566, −2.18909063071758120401657541575,
0.58581925314050236397475128310, 2.49434952479674914437258995962, 3.55306268458070136214567478093, 4.37989733842766822416819604378, 5.31351547866780890459760506994, 6.38704151982857577610552325109, 7.38900262807553939919463861798, 8.952513904638744612371294305448, 9.382029312410813661314205467618, 10.28607383812873043642123732784
