| L(s) = 1 | − 9.81i·3-s + (−16.9 − 53.2i)5-s − 222. i·7-s + 146.·9-s + 407.·11-s − 465. i·13-s + (−522. + 166. i)15-s + 284. i·17-s − 323.·19-s − 2.18e3·21-s + 12.0i·23-s + (−2.54e3 + 1.80e3i)25-s − 3.82e3i·27-s − 4.38e3·29-s + 1.01e4·31-s + ⋯ |
| L(s) = 1 | − 0.629i·3-s + (−0.303 − 0.952i)5-s − 1.71i·7-s + 0.603·9-s + 1.01·11-s − 0.764i·13-s + (−0.600 + 0.191i)15-s + 0.238i·17-s − 0.205·19-s − 1.08·21-s + 0.00475i·23-s + (−0.815 + 0.578i)25-s − 1.00i·27-s − 0.967·29-s + 1.89·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.775376955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.775376955\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (16.9 + 53.2i)T \) |
| good | 3 | \( 1 + 9.81iT - 243T^{2} \) |
| 7 | \( 1 + 222. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 407.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 465. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 284. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 323.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 12.0iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.01e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.90e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.00e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.50e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.05e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.58e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.86e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.10e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.31e4iT - 8.58e9T^{2} \) |
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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
−11.68431149561639764999896684328, −10.50110547913165215570426796989, −9.536479753814807006256316323068, −8.146349481201880462921081437166, −7.40269654795009979049489071249, −6.36096328196878131865429220386, −4.62037417186853709396838519029, −3.77539961007355453860398271868, −1.41428270642166964117653125424, −0.63932452637228997311136350731,
2.00408061202463107389115906174, 3.37724801904333355202847118174, 4.63380037944475450303829209652, 6.08134355460080770442200554943, 7.00815271344195603268909698422, 8.554080507590709462953834189900, 9.417342162472519049307748074580, 10.33271851978210846531842966003, 11.72376594881145591442040109471, 11.92643267376831994659623686925
