| L(s) = 1 | + (−0.984 + 0.173i)2-s + (1.71 − 0.233i)3-s + (0.939 − 0.342i)4-s + (0.0848 + 0.0712i)5-s + (−1.64 + 0.527i)6-s + (−2.61 + 0.405i)7-s + (−0.866 + 0.5i)8-s + (2.89 − 0.801i)9-s + (−0.0959 − 0.0554i)10-s + (3.36 + 4.00i)11-s + (1.53 − 0.806i)12-s + (3.05 + 0.538i)13-s + (2.50 − 0.853i)14-s + (0.162 + 0.102i)15-s + (0.766 − 0.642i)16-s + (−0.286 + 0.497i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.990 − 0.134i)3-s + (0.469 − 0.171i)4-s + (0.0379 + 0.0318i)5-s + (−0.673 + 0.215i)6-s + (−0.988 + 0.153i)7-s + (−0.306 + 0.176i)8-s + (0.963 − 0.267i)9-s + (−0.0303 − 0.0175i)10-s + (1.01 + 1.20i)11-s + (0.442 − 0.232i)12-s + (0.847 + 0.149i)13-s + (0.669 − 0.228i)14-s + (0.0419 + 0.0264i)15-s + (0.191 − 0.160i)16-s + (−0.0696 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37758 + 0.186039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37758 + 0.186039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-1.71 + 0.233i)T \) |
| 7 | \( 1 + (2.61 - 0.405i)T \) |
| good | 5 | \( 1 + (-0.0848 - 0.0712i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-3.36 - 4.00i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.05 - 0.538i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.286 - 0.497i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.21 + 4.16i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.79 - 4.93i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (8.47 - 1.49i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.29 + 6.30i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.699 - 1.21i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.875 + 4.96i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.33 - 4.47i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.79 - 2.83i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 6.95iT - 53T^{2} \) |
| 59 | \( 1 + (10.2 + 8.58i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.38 - 12.0i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.0706 - 0.400i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.50 + 3.75i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (12.2 - 7.08i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.16 + 6.63i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.04 - 5.94i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.86 + 6.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.16 + 6.15i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.36787053886820608872886074905, −10.04301165524780079432025058076, −9.298529081760008405828789788389, −9.041202654498540789369814673881, −7.52757205680173671581610196374, −7.06518923876991720637163410087, −5.92666405998144956507191085247, −4.11543315426268255549330191469, −3.00397721477575913068187965794, −1.55812665227429679233262098943,
1.33957958741471180955302763033, 3.22324566028245426109020501020, 3.66859879859613240040525503834, 5.73578124083377655722933584900, 6.81967066084462557354258537183, 7.75652008218835649520465414207, 8.879889834050359491875595490140, 9.229916785416362542753176821038, 10.20507859258155203546672805447, 11.10940696607559668877480633857
