L(s) = 1 | + 1.41i·2-s + (1.68 − 0.420i)3-s − 2.00·4-s + 3.91i·5-s + (0.595 + 2.37i)6-s − 2.64·7-s − 2.82i·8-s + (2.64 − 1.41i)9-s − 5.53·10-s + (−3.36 + 0.841i)12-s + 4.55·13-s − 3.74i·14-s + (1.64 + 6.57i)15-s + 4.00·16-s + (2 + 3.74i)18-s + 0.979·19-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + (0.970 − 0.242i)3-s − 1.00·4-s + 1.74i·5-s + (0.242 + 0.970i)6-s − 0.999·7-s − 1.00i·8-s + (0.881 − 0.471i)9-s − 1.74·10-s + (−0.970 + 0.242i)12-s + 1.26·13-s − 1.00i·14-s + (0.424 + 1.69i)15-s + 1.00·16-s + (0.471 + 0.881i)18-s + 0.224·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.812516 + 1.04114i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.812516 + 1.04114i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-1.68 + 0.420i)T \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 - 3.91iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 0.979T + 19T^{2} \) |
| 23 | \( 1 + 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 15.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 - 18.1iT - 83T^{2} \) |
| 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
−13.55082429441415591409002296235, −12.50221530797643592391357001073, −10.74265430773582338770900949019, −9.895123011417257168027292492298, −8.840629193448992531532894692301, −7.72884113914535507680162903545, −6.70838748321316437944352951414, −6.24246170687468710128333134418, −3.89135341580625531500095494698, −2.94718055445782401687889657671,
1.42638230889915777891905303173, 3.34275970871489480947168112588, 4.31870903147471137412630187115, 5.63570856612103873034045206863, 7.84834102385740257855322544008, 8.912533024591333633698434162696, 9.266482582456528696222940653048, 10.26744980315384565317696139653, 11.69523619649111839271029696122, 12.74811382195410214396024616967