| L(s) = 1 | + 2.05·2-s + 2.23·4-s − 5-s − 1.79·7-s + 0.473·8-s − 2.05·10-s + 0.0677·11-s + 2.70·13-s − 3.68·14-s − 3.48·16-s + 0.142·17-s − 2.32·19-s − 2.23·20-s + 0.139·22-s − 7.82·23-s + 25-s + 5.55·26-s − 3.99·28-s − 4.03·29-s − 2.01·31-s − 8.11·32-s + 0.292·34-s + 1.79·35-s − 4.70·37-s − 4.78·38-s − 0.473·40-s − 2.61·41-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.11·4-s − 0.447·5-s − 0.676·7-s + 0.167·8-s − 0.650·10-s + 0.0204·11-s + 0.749·13-s − 0.984·14-s − 0.871·16-s + 0.0345·17-s − 0.533·19-s − 0.498·20-s + 0.0297·22-s − 1.63·23-s + 0.200·25-s + 1.09·26-s − 0.754·28-s − 0.749·29-s − 0.361·31-s − 1.43·32-s + 0.0501·34-s + 0.302·35-s − 0.773·37-s − 0.776·38-s − 0.0747·40-s − 0.408·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 0.0677T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 - 0.142T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 + 2.01T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 + 0.0404T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 71 | \( 1 - 0.946T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 - 8.74T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
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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
−8.276350008805529385714788289990, −7.37468827498435084326928160675, −6.49708276678217562011886477240, −6.01297067817603986351197285997, −5.24721469820243595584158514150, −4.20958128698797899810927180739, −3.77468088053541606047489327958, −2.99398066775176224013142805752, −1.89029249716463410816007515361, 0,
1.89029249716463410816007515361, 2.99398066775176224013142805752, 3.77468088053541606047489327958, 4.20958128698797899810927180739, 5.24721469820243595584158514150, 6.01297067817603986351197285997, 6.49708276678217562011886477240, 7.37468827498435084326928160675, 8.276350008805529385714788289990
