L(s) = 1 | − 2.73·2-s + 1.63i·3-s + 5.47·4-s + (−0.903 − 2.04i)5-s − 4.46i·6-s + (1.05 − 1.05i)7-s − 9.51·8-s + 0.337·9-s + (2.46 + 5.59i)10-s + (1.15 + 1.15i)11-s + 8.94i·12-s + (1.53 − 1.53i)13-s + (−2.89 + 2.89i)14-s + (3.33 − 1.47i)15-s + 15.0·16-s + 7.16·17-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 0.942i·3-s + 2.73·4-s + (−0.403 − 0.914i)5-s − 1.82i·6-s + (0.400 − 0.400i)7-s − 3.36·8-s + 0.112·9-s + (0.781 + 1.76i)10-s + (0.348 + 0.348i)11-s + 2.58i·12-s + (0.424 − 0.424i)13-s + (−0.773 + 0.773i)14-s + (0.861 − 0.380i)15-s + 3.76·16-s + 1.73·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.517861 + 0.0640232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.517861 + 0.0640232i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.903 + 2.04i)T \) |
| 29 | \( 1 + (1.03 + 5.28i)T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 1.63iT - 3T^{2} \) |
| 7 | \( 1 + (-1.05 + 1.05i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.15 - 1.15i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.53 + 1.53i)T - 13iT^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 + (1.87 - 1.87i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.16 - 4.16i)T + 23iT^{2} \) |
| 31 | \( 1 + (0.504 + 0.504i)T + 31iT^{2} \) |
| 37 | \( 1 + 4.74iT - 37T^{2} \) |
| 41 | \( 1 + (-2.38 + 2.38i)T - 41iT^{2} \) |
| 43 | \( 1 + 8.05iT - 43T^{2} \) |
| 47 | \( 1 - 1.86iT - 47T^{2} \) |
| 53 | \( 1 + (5.39 + 5.39i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.57iT - 59T^{2} \) |
| 61 | \( 1 + (0.867 + 0.867i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.09 - 7.09i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.30iT - 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + (-2.01 + 2.01i)T - 79iT^{2} \) |
| 83 | \( 1 + (5.28 + 5.28i)T + 83iT^{2} \) |
| 89 | \( 1 + (10.1 - 10.1i)T - 89iT^{2} \) |
| 97 | \( 1 - 6.05iT - 97T^{2} \) |
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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
−12.66463043948580567544156706045, −11.64693255989155245920967652977, −10.70554875649521675137253639661, −9.848998034887321789359381316490, −9.168390323613494844989150491698, −8.086686288983822483544675215658, −7.34856872373015899847966260754, −5.54977714752280221174749831836, −3.74061710301242203305646874025, −1.27505063027461939698922179655,
1.36851071484157471660895127587, 2.92472401106778317220535264066, 6.22085297379968530752286681713, 7.00994057698612605081834699391, 7.84579012320735675108727229434, 8.687599388429043608740723848615, 9.930067753155201469242353237450, 10.93068934088555035963142894127, 11.66834163868817365135438899384, 12.55072336802118855374946002861