L(s) = 1 | − 1.89·2-s + 1.57·4-s − 2.79·5-s + 1.18·7-s + 0.806·8-s + 5.29·10-s − 6.19·11-s + 6.24·13-s − 2.24·14-s − 4.67·16-s − 7.04·17-s − 8.14·19-s − 4.40·20-s + 11.7·22-s + 23-s + 2.83·25-s − 11.8·26-s + 1.86·28-s − 29-s − 1.41·31-s + 7.21·32-s + 13.3·34-s − 3.32·35-s + 1.97·37-s + 15.3·38-s − 2.25·40-s − 1.66·41-s + ⋯ |
L(s) = 1 | − 1.33·2-s + 0.786·4-s − 1.25·5-s + 0.448·7-s + 0.285·8-s + 1.67·10-s − 1.86·11-s + 1.73·13-s − 0.599·14-s − 1.16·16-s − 1.70·17-s − 1.86·19-s − 0.984·20-s + 2.49·22-s + 0.208·23-s + 0.566·25-s − 2.31·26-s + 0.353·28-s − 0.185·29-s − 0.253·31-s + 1.27·32-s + 2.28·34-s − 0.561·35-s + 0.324·37-s + 2.49·38-s − 0.356·40-s − 0.259·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1480257732\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1480257732\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 1.97T + 37T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 + 1.54T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 8.99T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 4.61T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 0.494T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 4.20T + 97T^{2} \) |
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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
−8.288595936629777134294613239292, −7.80232044226008030745103677246, −6.86624987282502813038156760319, −6.31600909335543340818581004984, −5.04717777751267638708017533918, −4.42430388673341389771203105141, −3.68024947841093300459842061381, −2.48386193656554289834426792352, −1.64902148312435331015754005772, −0.24415521935247141333982975998,
0.24415521935247141333982975998, 1.64902148312435331015754005772, 2.48386193656554289834426792352, 3.68024947841093300459842061381, 4.42430388673341389771203105141, 5.04717777751267638708017533918, 6.31600909335543340818581004984, 6.86624987282502813038156760319, 7.80232044226008030745103677246, 8.288595936629777134294613239292