| L(s) = 1 | + 1.16i·2-s + 0.645·4-s − 2.64·7-s + 3.07i·8-s − 6.57i·11-s − 3.07i·14-s − 2.29·16-s + 7.64·22-s + 1.91i·23-s − 5·25-s − 1.70·28-s + 8.89i·29-s + 3.49i·32-s + 10.5·37-s − 5.29·43-s − 4.24i·44-s + ⋯ |
| L(s) = 1 | + 0.822i·2-s + 0.322·4-s − 0.999·7-s + 1.08i·8-s − 1.98i·11-s − 0.822i·14-s − 0.572·16-s + 1.63·22-s + 0.399i·23-s − 25-s − 0.322·28-s + 1.65i·29-s + 0.617i·32-s + 1.73·37-s − 0.806·43-s − 0.639i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.821135 + 0.425051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.821135 + 0.425051i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 2 | \( 1 - 1.16iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.57iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.91iT - 23T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 0.412iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 15.0iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−15.37839617751811638574640413507, −14.15100869054803846173209814490, −13.22009549831164204490559065357, −11.71460775244611096458737915195, −10.70075553940804794258546210474, −9.102503204622770221322359963046, −7.914598166234394637787708660489, −6.51704390189487830140740976882, −5.65581644885609870537574479063, −3.21747895858856631008707763599,
2.36910700508465272032724899456, 4.14866449520297582156615506030, 6.34971266648369876382104934076, 7.49854097002460438643128936049, 9.633496778371731059670292825705, 10.07756726314679396730993129567, 11.58928739925306627400064380149, 12.48276671159758702308905729504, 13.27395022401112660829738901090, 15.01011981222575095933958724690
