| L(s) = 1 | + (0.808 − 0.359i)2-s + (−0.814 + 0.904i)4-s + (−1.09 − 1.94i)5-s + (−0.458 + 0.793i)7-s + (−0.879 + 2.70i)8-s + (−1.58 − 1.18i)10-s + (−1.33 + 0.594i)11-s + (−1.67 − 0.744i)13-s + (−0.0846 + 0.805i)14-s + (0.00858 + 0.0816i)16-s + (−1.50 + 4.62i)17-s + (−2.04 + 6.28i)19-s + (2.65 + 0.596i)20-s + (−0.864 + 0.960i)22-s + (0.0683 − 0.650i)23-s + ⋯ |
| L(s) = 1 | + (0.571 − 0.254i)2-s + (−0.407 + 0.452i)4-s + (−0.490 − 0.871i)5-s + (−0.173 + 0.299i)7-s + (−0.310 + 0.956i)8-s + (−0.501 − 0.373i)10-s + (−0.402 + 0.179i)11-s + (−0.464 − 0.206i)13-s + (−0.0226 + 0.215i)14-s + (0.00214 + 0.0204i)16-s + (−0.364 + 1.12i)17-s + (−0.468 + 1.44i)19-s + (0.594 + 0.133i)20-s + (−0.184 + 0.204i)22-s + (0.0142 − 0.135i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.283107 + 0.559376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.283107 + 0.559376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.09 + 1.94i)T \) |
| good | 2 | \( 1 + (-0.808 + 0.359i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (0.458 - 0.793i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.33 - 0.594i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (1.67 + 0.744i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (1.50 - 4.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.04 - 6.28i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0683 + 0.650i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (6.29 + 1.33i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-2.87 + 0.611i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.86 - 3.53i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.26 + 1.00i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 3.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.26 + 1.11i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-3.46 - 10.6i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.82 + 0.813i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-11.5 + 5.14i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-8.34 + 1.77i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (5.09 + 15.6i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.01 + 1.46i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (6.05 + 1.28i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (2.67 + 2.97i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-11.4 - 8.29i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.89 + 1.25i)T + (88.6 + 39.4i)T^{2} \) |
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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
−10.94297027577048786345552974656, −9.928879635779083682743889045339, −8.898362196699378792734001096454, −8.251714395785708142166477852290, −7.58021253283423681868987909111, −6.03165383603786442712131005230, −5.18013338364048408665384942556, −4.25176603218733366167394972776, −3.48584192296013198536627979454, −2.01912289369378896949316532191,
0.26600213075620108105628774570, 2.56743629037453570890440064579, 3.70909842262759412145938970602, 4.68591684782466193230704517139, 5.55429196166883313734353125342, 6.92598155428228997477820285460, 7.01659944871818698553464086547, 8.468446617288245677137249918745, 9.469211551327045894396065164894, 10.19136026835284002099652291814
