L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + (−2.5 − 4.33i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)11-s − 0.999·12-s + (2.5 − 2.59i)13-s + 5·14-s + (−0.5 + 0.866i)16-s + (−4 − 6.92i)17-s + 0.999·18-s + (2.5 + 4.33i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s + (−0.944 − 1.63i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.452 − 0.783i)11-s − 0.288·12-s + (0.693 − 0.720i)13-s + 1.33·14-s + (−0.125 + 0.216i)16-s + (−0.970 − 1.68i)17-s + 0.235·18-s + (0.573 + 0.993i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8817661334\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8817661334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
good | 7 | \( 1 + (2.5 + 4.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1 + 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-5.5 + 9.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-48.5 + 84.0i)T^{2} \) |
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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
−8.782708619609935481433878211102, −7.83540007868342819771502571880, −7.41971767222240864084276215627, −6.50574419344820441035150218202, −6.11495133337382610329286559508, −4.84756799930984082284646857199, −3.71767503134361079224763046582, −3.10113767207549748088282599526, −1.28770776188150996851293617787, −0.36417996033996846818992111353,
1.86280931083554841523579625705, 2.54627793161677866062291266947, 3.63152583939253463928614654608, 4.34516099912894322106774768519, 5.49887297664766994310760049203, 6.35288119794000054171307021912, 7.11331747970344182469529886768, 8.556830018957462287171729480271, 8.872802615036259841930815296848, 9.282476357624505982902813602347