L(s) = 1 | − 2-s + (0.787 − 1.54i)3-s + 4-s + (−3.27 − 1.89i)5-s + (−0.787 + 1.54i)6-s + (0.475 + 2.60i)7-s − 8-s + (−1.76 − 2.42i)9-s + (3.27 + 1.89i)10-s + (−2.90 + 5.03i)11-s + (0.787 − 1.54i)12-s + (−0.879 + 3.49i)13-s + (−0.475 − 2.60i)14-s + (−5.50 + 3.56i)15-s + 16-s + 4.53·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.454 − 0.890i)3-s + 0.5·4-s + (−1.46 − 0.846i)5-s + (−0.321 + 0.629i)6-s + (0.179 + 0.983i)7-s − 0.353·8-s + (−0.586 − 0.809i)9-s + (1.03 + 0.598i)10-s + (−0.875 + 1.51i)11-s + (0.227 − 0.445i)12-s + (−0.243 + 0.969i)13-s + (−0.126 − 0.695i)14-s + (−1.42 + 0.921i)15-s + 0.250·16-s + 1.10·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.330770 + 0.283484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.330770 + 0.283484i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.787 + 1.54i)T \) |
| 7 | \( 1 + (-0.475 - 2.60i)T \) |
| 13 | \( 1 + (0.879 - 3.49i)T \) |
good | 5 | \( 1 + (3.27 + 1.89i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.90 - 5.03i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 + (1.42 + 2.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.84iT - 23T^{2} \) |
| 29 | \( 1 + (-2.20 + 1.27i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.824 + 1.42i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.999iT - 37T^{2} \) |
| 41 | \( 1 + (-3.48 + 2.01i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.445 - 0.772i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.23 + 3.59i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.41 - 3.12i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 10.0iT - 59T^{2} \) |
| 61 | \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.24 + 5.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.77 - 11.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.19 - 3.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.40 - 14.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 2.18iT - 89T^{2} \) |
| 97 | \( 1 + (-6.30 + 10.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.31767670649421141258756141538, −9.765567801185357652064079282516, −9.045727193485396958802629746494, −8.216913716520022804927843668173, −7.63216255781217130437509831424, −6.98262324486034918831135247384, −5.47603932548345764969483431964, −4.32794280704198810964482470757, −2.80395744338280181429019219841, −1.57149575199092530779589812068,
0.30048853884473995321285799953, 3.09165245338908250182386174181, 3.40420909041097684336417896442, 4.73287913135303585237060661596, 6.12196684898829301744057075236, 7.58591891049918860880816567599, 7.924763197749805692392028392731, 8.554754660240830908813349575857, 10.03667889892999888396914866483, 10.70373636181347870408696068473