| L(s) = 1 | − 4.43i·2-s − 3.69·4-s − 22.3i·5-s + 51.9i·7-s − 54.6i·8-s − 99.2·10-s + 25.0·11-s − 38.7·13-s + 230.·14-s − 301.·16-s − 111.·17-s − 238. i·19-s + 82.5i·20-s − 111. i·22-s − 823.·23-s + ⋯ |
| L(s) = 1 | − 1.10i·2-s − 0.230·4-s − 0.894i·5-s + 1.06i·7-s − 0.853i·8-s − 0.992·10-s + 0.206·11-s − 0.229·13-s + 1.17·14-s − 1.17·16-s − 0.385·17-s − 0.661i·19-s + 0.206i·20-s − 0.229i·22-s − 1.55·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5155892429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5155892429\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (528. + 1.77e3i)T \) |
| good | 2 | \( 1 + 4.43iT - 16T^{2} \) |
| 5 | \( 1 + 22.3iT - 625T^{2} \) |
| 7 | \( 1 - 51.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 25.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + 38.7T + 2.85e4T^{2} \) |
| 17 | \( 1 + 111.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 238. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 823.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 424. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.44e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 626. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 580.T + 2.82e6T^{2} \) |
| 47 | \( 1 - 170.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.30e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 65.4T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.89e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.44e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 5.84e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.77e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.01e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.43e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 9.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 7.76e3T + 8.85e7T^{2} \) |
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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
−10.12834282822632143924240725319, −9.170565004528820355352309947846, −8.674708270336736269891988004639, −7.26574584152469804167667395948, −6.02872219302547829266343558882, −4.94677034410655703411504276555, −3.80006231064472781982551672661, −2.49880309908621744402782337666, −1.58764373160799065391917418085, −0.13238644615401239651160241918,
1.93026701749086385180003894651, 3.43743636943665563082382121536, 4.63505203798536152917191352957, 5.99730215890663672123518115093, 6.66564504621778551676932905424, 7.50363365050789977951340486161, 8.109891115940023039538401280152, 9.480998586943272225089952769204, 10.50972549384811931763065073141, 11.12572600215304912967194112704
