| L(s) = 1 | + (−4 + 3.31i)3-s − 8·4-s + 13.2i·5-s + (5 − 26.5i)9-s + 36.4i·11-s + (32 − 26.5i)12-s + (−44 − 53.0i)15-s + 64·16-s − 106. i·20-s + 192. i·23-s − 51·25-s + (68 + 122. i)27-s − 340·31-s + (−121 − 145. i)33-s + (−40 + 212. i)36-s + 434·37-s + ⋯ |
| L(s) = 1 | + (−0.769 + 0.638i)3-s − 4-s + 1.18i·5-s + (0.185 − 0.982i)9-s + 1.00i·11-s + (0.769 − 0.638i)12-s + (−0.757 − 0.913i)15-s + 16-s − 1.18i·20-s + 1.74i·23-s − 0.408·25-s + (0.484 + 0.874i)27-s − 1.96·31-s + (−0.638 − 0.769i)33-s + (−0.185 + 0.982i)36-s + 1.92·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.272241 + 0.579382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272241 + 0.579382i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4 - 3.31i)T \) |
| 11 | \( 1 - 36.4iT \) |
| good | 2 | \( 1 + 8T^{2} \) |
| 5 | \( 1 - 13.2iT - 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 - 192. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 340T + 2.97e4T^{2} \) |
| 37 | \( 1 - 434T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 643. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 225. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 550. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 416T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 - 132. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 34T + 9.12e5T^{2} \) |
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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
−16.84554703144058556385728820735, −15.26261514956938715094399287408, −14.53693040595190469056042496956, −13.01567201937946566267444944278, −11.54873727461745405209685470654, −10.29921769304963799277093712860, −9.370413656075901871775726114474, −7.22695323575738714199770440192, −5.49093125966239897744959337270, −3.84731316563188923300606049744,
0.68018919522407504636672132604, 4.61241614486548268154450395183, 5.88253718205952308588782110229, 8.035652499512990361122647576289, 9.142613667290219725271417935311, 10.90826767119596348071720584485, 12.48522507971855363441782072949, 13.06046196518666073324503886292, 14.27087048812727286956085780821, 16.37553697149197884722126398811
