| L(s) = 1 | + (−3.49 − 3.49i)2-s − 7.53i·4-s + (−24.1 + 24.1i)7-s + (−138. + 138. i)8-s − 173. i·11-s + (−597. − 597. i)13-s + 168.·14-s + 726.·16-s + (734. + 734. i)17-s + 1.62e3i·19-s + (−607. + 607. i)22-s + (−667. + 667. i)23-s + 4.18e3i·26-s + (181. + 181. i)28-s + 3.69e3·29-s + ⋯ |
| L(s) = 1 | + (−0.618 − 0.618i)2-s − 0.235i·4-s + (−0.186 + 0.186i)7-s + (−0.763 + 0.763i)8-s − 0.432i·11-s + (−0.981 − 0.981i)13-s + 0.230·14-s + 0.709·16-s + (0.616 + 0.616i)17-s + 1.03i·19-s + (−0.267 + 0.267i)22-s + (−0.262 + 0.262i)23-s + 1.21i·26-s + (0.0438 + 0.0438i)28-s + 0.816·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9077046868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9077046868\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (3.49 + 3.49i)T + 32iT^{2} \) |
| 7 | \( 1 + (24.1 - 24.1i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 173. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (597. + 597. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-734. - 734. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.62e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (667. - 667. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 3.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-5.75e3 + 5.75e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.60e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (9.78e3 + 9.78e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.11e4 - 1.11e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.01e3 + 1.01e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 4.65e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.12e4 + 3.12e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.24e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.58e3 - 3.58e3i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 1.75e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.11e4 + 7.11e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 4.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-7.90e3 + 7.90e3i)T - 8.58e9iT^{2} \) |
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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
−11.18925639434139593905035816622, −10.25214978833206390633079112070, −9.702175264297140345323488946620, −8.552783281835712765356124955704, −7.65308091437153432766596694432, −6.05655461886829093517591056925, −5.28347943545306524240961144253, −3.47638494573753682791475774857, −2.23058732409056493324279429580, −0.865643359389463604702150543070,
0.43949497476288645936280076659, 2.42503209708944825350850778516, 3.88196581721253656187510670388, 5.20812390647451093681064667527, 6.84085064438286159825206247355, 7.17647373993778670738726935035, 8.401137568730374689832567434468, 9.361020729216800243216319091724, 10.02143288586369625949828738902, 11.50153169526348041092323484371
