| L(s) = 1 | + (−10.9 + 11.0i)3-s + 30.7·5-s + (−2.58 − 242. i)9-s + 13.8i·11-s − 860. i·13-s + (−337. + 341. i)15-s − 1.17e3·17-s + 681. i·19-s − 880. i·23-s − 2.17e3·25-s + (2.72e3 + 2.63e3i)27-s − 3.78e3i·29-s + 6.09e3i·31-s + (−153. − 152. i)33-s − 2.43e3·37-s + ⋯ |
| L(s) = 1 | + (−0.703 + 0.710i)3-s + 0.550·5-s + (−0.0106 − 0.999i)9-s + 0.0346i·11-s − 1.41i·13-s + (−0.387 + 0.391i)15-s − 0.984·17-s + 0.433i·19-s − 0.347i·23-s − 0.696·25-s + (0.718 + 0.695i)27-s − 0.835i·29-s + 1.13i·31-s + (−0.0246 − 0.0243i)33-s − 0.292·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8609393912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8609393912\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (10.9 - 11.0i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 30.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 13.8iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 860. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 681. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 880. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.78e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.09e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.74e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.25e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 5.68e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.59e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.75e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 5.62e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.55e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.12e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.94e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.37e4iT - 8.58e9T^{2} \) |
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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
−10.39607371624377691932867078697, −9.500268123002302675340877659598, −8.659176601540254237672932818793, −7.52137354423008206772920795680, −6.30472666669312363207989400921, −5.69599189947823478982020106523, −4.76667590404267728668928568506, −3.72213364001292556529880365595, −2.49455562517168638319220624221, −0.946234843916634256259961719873,
0.23923915975400780849660285361, 1.62505323595414221299209749344, 2.36153583604324585569624415056, 4.10417057525675308202396111787, 5.11176450729189502917827738409, 6.11643343598016610781745979895, 6.77511812525332803045632315781, 7.61497253753000996848054084092, 8.815910473928303456507536742331, 9.556203968361779455244994979674
