| L(s) = 1 | + (−2.5 − 1.65i)3-s + 3.31·5-s − 7i·7-s + (3.5 + 8.29i)9-s − 19.8·11-s − 13i·13-s + (−8.29 − 5.5i)15-s − 23.2i·17-s − 30i·19-s + (−11.6 + 17.5i)21-s + 26.5i·23-s − 14·25-s + (4.99 − 26.5i)27-s + 33.1i·29-s − 10i·31-s + ⋯ |
| L(s) = 1 | + (−0.833 − 0.552i)3-s + 0.663·5-s − i·7-s + (0.388 + 0.921i)9-s − 1.80·11-s − i·13-s + (−0.552 − 0.366i)15-s − 1.36i·17-s − 1.57i·19-s + (−0.552 + 0.833i)21-s + 1.15i·23-s − 0.560·25-s + (0.185 − 0.982i)27-s + 1.14i·29-s − 0.322i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.415008 - 0.773295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415008 - 0.773295i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.5 + 1.65i)T \) |
| 13 | \( 1 + 13iT \) |
| good | 5 | \( 1 - 3.31T + 25T^{2} \) |
| 7 | \( 1 + 7iT - 49T^{2} \) |
| 11 | \( 1 + 19.8T + 121T^{2} \) |
| 17 | \( 1 + 23.2iT - 289T^{2} \) |
| 19 | \( 1 + 30iT - 361T^{2} \) |
| 23 | \( 1 - 26.5iT - 529T^{2} \) |
| 29 | \( 1 - 33.1iT - 841T^{2} \) |
| 31 | \( 1 + 10iT - 961T^{2} \) |
| 37 | \( 1 - 13iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 55T + 1.84e3T^{2} \) |
| 47 | \( 1 - 49.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 39.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 92.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 14T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 29.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 54T + 6.24e3T^{2} \) |
| 83 | \( 1 - 33.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 19.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 78iT - 9.40e3T^{2} \) |
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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
−12.55437651439348987486740093466, −11.11181255810964703944207608157, −10.60004865659824332702336286214, −9.540215439167404283204077465927, −7.72517547592097234059331833595, −7.18804205827608915971351313686, −5.66371336293876474351067653663, −4.91431211210998471922237647992, −2.65533525078992048287167881540, −0.59402013560230278953017744788,
2.24687389841936803684487457584, 4.25205120928399733494593069769, 5.68309921313460588910684234404, 6.08136923529200378577822445977, 7.915105774694383149082892751279, 9.148901320295179468709797098845, 10.19903470216553760875536084508, 10.82579923724576719367586123944, 12.19694254188398388049068271708, 12.70737307571162085427735275109
