| L(s) = 1 | + 22.6·2-s + (−18.8 + 242. i)3-s + 512.·4-s + (−426. + 5.48e3i)6-s − 670. i·7-s + 1.15e4·8-s + (−5.83e4 − 9.12e3i)9-s − 2.33e5i·11-s + (−9.64e3 + 1.24e5i)12-s + 3.07e5i·13-s − 1.51e4i·14-s + 2.62e5·16-s − 6.72e5·17-s + (−1.32e6 − 2.06e5i)18-s + 1.55e6·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.0775 + 0.996i)3-s + 0.500·4-s + (−0.0548 + 0.704i)6-s − 0.0398i·7-s + 0.353·8-s + (−0.987 − 0.154i)9-s − 1.44i·11-s + (−0.0387 + 0.498i)12-s + 0.828i·13-s − 0.0282i·14-s + 0.250·16-s − 0.473·17-s + (−0.698 − 0.109i)18-s + 0.626·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.326104291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.326104291\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 22.6T \) |
| 3 | \( 1 + (18.8 - 242. i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 670. iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.33e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 3.07e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 6.72e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.55e6T + 6.13e12T^{2} \) |
| 23 | \( 1 - 5.57e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + 2.97e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 3.09e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 8.56e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 3.59e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 3.66e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 3.28e7T + 5.25e16T^{2} \) |
| 53 | \( 1 - 4.59e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 4.88e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 6.12e7T + 7.13e17T^{2} \) |
| 67 | \( 1 - 6.70e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.23e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.08e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 1.86e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.09e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 5.19e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 1.07e10iT - 7.37e19T^{2} \) |
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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
−11.33906009068946334382292646238, −10.34083535727240584603936906468, −9.200355216488540868421126246649, −8.219455441094619854787360428367, −6.60953404294973343039053941104, −5.65417090967375605150420952232, −4.59458653568935470928182195416, −3.60481549083769462291188564087, −2.59083109891444334535273636167, −0.73145358007294026746120474456,
0.895370877168620173456499341852, 2.07312220039832839593369890261, 3.10695673882521244508692296655, 4.69779731081993292469952284759, 5.68651406985824449541221074523, 6.93243527379791225864219521969, 7.51468012578222364041254145909, 8.841950814716166869572897453116, 10.26045397905513689534633789745, 11.35015171527610300991418595349
