L(s) = 1 | + (−0.842 − 1.45i)5-s + (2.30 + 1.30i)7-s + (−3.38 − 1.95i)11-s − 6.05i·13-s + (−0.201 + 0.348i)17-s + (−0.145 + 0.0840i)19-s + (−7.69 + 4.44i)23-s + (1.07 − 1.86i)25-s + 7.10i·29-s + (−5.44 − 3.14i)31-s + (−0.0398 − 4.45i)35-s + (3.13 + 5.42i)37-s − 3.29·41-s − 3.60·43-s + (−4.38 − 7.59i)47-s + ⋯ |
L(s) = 1 | + (−0.376 − 0.652i)5-s + (0.870 + 0.492i)7-s + (−1.01 − 0.588i)11-s − 1.67i·13-s + (−0.0488 + 0.0845i)17-s + (−0.0334 + 0.0192i)19-s + (−1.60 + 0.926i)23-s + (0.215 − 0.373i)25-s + 1.31i·29-s + (−0.977 − 0.564i)31-s + (−0.00673 − 0.753i)35-s + (0.514 + 0.891i)37-s − 0.514·41-s − 0.550·43-s + (−0.639 − 1.10i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4912400111\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4912400111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.30 - 1.30i)T \) |
good | 5 | \( 1 + (0.842 + 1.45i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.05iT - 13T^{2} \) |
| 17 | \( 1 + (0.201 - 0.348i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.145 - 0.0840i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.69 - 4.44i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.10iT - 29T^{2} \) |
| 31 | \( 1 + (5.44 + 3.14i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 3.60T + 43T^{2} \) |
| 47 | \( 1 + (4.38 + 7.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.94 - 2.85i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.25 - 3.89i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.43 - 2.56i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.95 + 5.11i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (6.05 + 3.49i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.603 + 1.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.362T + 83T^{2} \) |
| 89 | \( 1 + (1.38 + 2.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.587iT - 97T^{2} \) |
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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
−8.301501122675505940775781685980, −8.146940779861641876086522836605, −7.43329018136293206922119363763, −6.02403983100956513332583586886, −5.38885408238027551755078710755, −4.87240380981019849455935874342, −3.68880774058431601013667691937, −2.76892131441237639643734427257, −1.55199985106914609339489991173, −0.16028245909201567507987976487,
1.75114776372090553859877468036, 2.52252130156810448666679432007, 3.93157130481422421640835373004, 4.41430048100275552012322936219, 5.35923350492801453524759641205, 6.47160978873321350154991472318, 7.14514889866415689838727609745, 7.79266780700399526979602998025, 8.443154983307577018357475794339, 9.473865714467932496805532087170