| L(s) = 1 | + (−1.81 + 1.12i)2-s + (0.273 + 0.961i)3-s + (1.14 − 2.29i)4-s + (−2.35 + 0.913i)5-s + (−1.57 − 1.43i)6-s + (−0.431 + 4.65i)7-s + (0.111 + 1.20i)8-s + (−0.850 + 0.526i)9-s + (3.25 − 4.31i)10-s + (0.992 − 0.614i)11-s + (2.52 + 0.471i)12-s + (0.222 − 2.40i)13-s + (−4.45 − 8.94i)14-s + (−1.52 − 2.01i)15-s + (1.53 + 2.02i)16-s + (−5.21 + 4.75i)17-s + ⋯ |
| L(s) = 1 | + (−1.28 + 0.795i)2-s + (0.157 + 0.555i)3-s + (0.572 − 1.14i)4-s + (−1.05 + 0.408i)5-s + (−0.644 − 0.587i)6-s + (−0.163 + 1.76i)7-s + (0.0395 + 0.426i)8-s + (−0.283 + 0.175i)9-s + (1.02 − 1.36i)10-s + (0.299 − 0.185i)11-s + (0.728 + 0.136i)12-s + (0.0617 − 0.666i)13-s + (−1.19 − 2.39i)14-s + (−0.393 − 0.520i)15-s + (0.383 + 0.507i)16-s + (−1.26 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.128036 - 0.205663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128036 - 0.205663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.273 - 0.961i)T \) |
| 103 | \( 1 + (-6.01 - 8.17i)T \) |
| good | 2 | \( 1 + (1.81 - 1.12i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (2.35 - 0.913i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.431 - 4.65i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (-0.992 + 0.614i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.222 + 2.40i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (5.21 - 4.75i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.29 + 4.53i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (5.59 + 3.46i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-7.38 + 2.86i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (2.92 + 3.87i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-1.22 - 0.229i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-6.68 - 2.58i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (6.73 - 1.25i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 1.85T + 47T^{2} \) |
| 53 | \( 1 + (-0.377 + 1.32i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (-0.942 - 10.1i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (5.19 - 4.73i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-1.20 - 13.0i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (4.27 + 1.65i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-1.49 - 0.579i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (11.1 - 4.32i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (-0.199 + 2.15i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-7.01 - 14.0i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (4.05 + 3.69i)T + (8.95 + 96.5i)T^{2} \) |
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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
−12.00013858835102391723370805759, −11.18804488847666104123318321075, −10.21005878074929753814969497040, −9.178477186783451486999266786998, −8.512077454829293167588816995643, −7.973771881925090935193377074635, −6.61056856198421382609888236600, −5.81623556270910404708567435446, −4.16982727339138699746903823658, −2.64019888908349777515547113137,
0.26451975040935556703139999887, 1.59878703467455913729999426014, 3.45626327656676748370551703696, 4.51234624997929670899680123423, 6.76062239613236810919470293596, 7.53125804893702786234532892468, 8.197749750197395523042445997486, 9.234025315043501077254604295334, 10.11346835212506632172787937372, 11.06474984172099457873731478902
