| L(s) = 1 | + (−15.3 + 12.8i)3-s + (−87.8 − 31.9i)5-s + (31.3 − 54.2i)7-s + (27.3 − 155. i)9-s + (−93.1 − 161. i)11-s + (738. + 619. i)13-s + (1.75e3 − 639. i)15-s + (71.8 + 407. i)17-s + (−1.22e3 + 985. i)19-s + (217. + 1.23e3i)21-s + (3.92e3 − 1.42e3i)23-s + (4.29e3 + 3.60e3i)25-s + (−853. − 1.47e3i)27-s + (1.45e3 − 8.27e3i)29-s + (625. − 1.08e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.983 + 0.825i)3-s + (−1.57 − 0.571i)5-s + (0.241 − 0.418i)7-s + (0.112 − 0.639i)9-s + (−0.232 − 0.402i)11-s + (1.21 + 1.01i)13-s + (2.01 − 0.734i)15-s + (0.0603 + 0.341i)17-s + (−0.779 + 0.626i)19-s + (0.107 + 0.611i)21-s + (1.54 − 0.563i)23-s + (1.37 + 1.15i)25-s + (−0.225 − 0.390i)27-s + (0.322 − 1.82i)29-s + (0.116 − 0.202i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00566i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.809984 - 0.00229596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.809984 - 0.00229596i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.22e3 - 985. i)T \) |
| good | 3 | \( 1 + (15.3 - 12.8i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (87.8 + 31.9i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-31.3 + 54.2i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (93.1 + 161. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-738. - 619. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-71.8 - 407. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-3.92e3 + 1.42e3i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-1.45e3 + 8.27e3i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-625. + 1.08e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 5.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.16e3 + 7.69e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (9.27e3 + 3.37e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (1.46e3 - 8.29e3i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-9.23e3 + 3.35e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-2.08e3 - 1.18e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (2.58e4 - 9.40e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (3.50e3 - 1.98e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (1.79e4 + 6.54e3i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-3.33e4 + 2.79e4i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-4.79e4 + 4.02e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-3.18e4 + 5.52e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-6.88e4 - 5.77e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (9.58e3 + 5.43e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−13.39226348629178655854179241877, −12.05559230204862744236461666136, −11.24728485809224326889347221284, −10.62849374193839538238047915322, −8.888609547706434153158908901229, −7.84083513045891407707041007839, −6.18234001238583205053933115609, −4.57873461334432346467362388350, −3.93983702708602098561592342403, −0.64073169165292147905516546055,
0.830298805310458569957791109473, 3.25885971714822909815908792960, 5.06945424448968664937594798573, 6.58177385741506403485907092676, 7.48440678841379107329807495693, 8.629668139729402320688897774748, 10.86707090801096358071312252645, 11.26910397500799640332354855539, 12.34496615250627704345296343588, 13.10535440234241602953558057299
