L(s) = 1 | + (−1.27 − 0.601i)2-s + (1.55 + 1.55i)3-s + (1.27 + 1.54i)4-s + (1.95 − 1.08i)5-s + (−1.05 − 2.92i)6-s + (3.02 + 3.02i)7-s + (−0.706 − 2.73i)8-s + 1.82i·9-s + (−3.15 + 0.205i)10-s − 4.00·11-s + (−0.409 + 4.37i)12-s + (−0.933 − 3.48i)13-s + (−2.05 − 5.70i)14-s + (4.71 + 1.36i)15-s + (−0.743 + 3.93i)16-s + (−0.753 − 0.753i)17-s + ⋯ |
L(s) = 1 | + (−0.904 − 0.425i)2-s + (0.896 + 0.896i)3-s + (0.638 + 0.770i)4-s + (0.875 − 0.483i)5-s + (−0.429 − 1.19i)6-s + (1.14 + 1.14i)7-s + (−0.249 − 0.968i)8-s + 0.606i·9-s + (−0.997 + 0.0651i)10-s − 1.20·11-s + (−0.118 + 1.26i)12-s + (−0.258 − 0.965i)13-s + (−0.549 − 1.52i)14-s + (1.21 + 0.351i)15-s + (−0.185 + 0.982i)16-s + (−0.182 − 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28233 + 0.287593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28233 + 0.287593i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.27 + 0.601i)T \) |
| 5 | \( 1 + (-1.95 + 1.08i)T \) |
| 13 | \( 1 + (0.933 + 3.48i)T \) |
good | 3 | \( 1 + (-1.55 - 1.55i)T + 3iT^{2} \) |
| 7 | \( 1 + (-3.02 - 3.02i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 17 | \( 1 + (0.753 + 0.753i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.01iT - 19T^{2} \) |
| 23 | \( 1 + (3.75 + 3.75i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.60iT - 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 + (0.454 - 0.454i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.01iT - 41T^{2} \) |
| 43 | \( 1 + (4.32 + 4.32i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.87 - 3.87i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.34 + 1.34i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.838iT - 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + (3.69 + 3.69i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + (-5.33 - 5.33i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.43T + 79T^{2} \) |
| 83 | \( 1 + (-4.47 + 4.47i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.30T + 89T^{2} \) |
| 97 | \( 1 + (-9.64 + 9.64i)T - 97iT^{2} \) |
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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
−12.04749267379940251657829326551, −10.54534228937755372560072934102, −10.19781293643616045592718488159, −9.055083650863794571591616572327, −8.510477064169096786927919422144, −7.81390945796863846274443032908, −5.81822630676286412818251083800, −4.75373170673852127244828492348, −2.97400272712245451568766907795, −2.03808965392176013663625205666,
1.59656620099104308174635813733, 2.52388844626297441385722391261, 4.85821632942325936159305479294, 6.33685360272056698894649282631, 7.42114362325684695397322100056, 7.75923971261173218480444515494, 8.836927574841579725904585125691, 9.948003619910104242824346636544, 10.73265494077961190385683872745, 11.62922360246540092783000792701