| L(s) = 1 | + (1.25 − 1.04i)2-s + (−0.905 − 1.47i)3-s + (0.115 − 0.656i)4-s + (0.939 − 0.342i)5-s + (−2.68 − 0.895i)6-s + (−0.576 − 3.26i)7-s + (1.08 + 1.88i)8-s + (−1.35 + 2.67i)9-s + (0.816 − 1.41i)10-s + (−3.04 − 1.10i)11-s + (−1.07 + 0.423i)12-s + (4.99 + 4.19i)13-s + (−4.15 − 3.48i)14-s + (−1.35 − 1.07i)15-s + (4.59 + 1.67i)16-s + (−0.561 + 0.972i)17-s + ⋯ |
| L(s) = 1 | + (0.884 − 0.742i)2-s + (−0.522 − 0.852i)3-s + (0.0579 − 0.328i)4-s + (0.420 − 0.152i)5-s + (−1.09 − 0.365i)6-s + (−0.217 − 1.23i)7-s + (0.384 + 0.666i)8-s + (−0.452 + 0.891i)9-s + (0.258 − 0.447i)10-s + (−0.918 − 0.334i)11-s + (−0.310 + 0.122i)12-s + (1.38 + 1.16i)13-s + (−1.10 − 0.931i)14-s + (−0.350 − 0.278i)15-s + (1.14 + 0.418i)16-s + (−0.136 + 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0360 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0360 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01539 - 1.05272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01539 - 1.05272i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.905 + 1.47i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| good | 2 | \( 1 + (-1.25 + 1.04i)T + (0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (0.576 + 3.26i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.04 + 1.10i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.99 - 4.19i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.561 - 0.972i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0708 - 0.122i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0812 - 0.460i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.61 - 1.35i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.58 - 8.98i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.42 + 7.90i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.68 + 0.975i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.226 - 1.28i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + (4.34 - 1.57i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.313 - 1.78i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.22 + 7.73i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.293 + 0.507i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 1.99i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.47 + 7.11i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.76 + 1.48i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.01 - 5.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.81 - 3.57i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.10726999270392342717615826982, −12.15491764663851019632128591789, −10.97145953524093869752452208048, −10.61056513189269779717603616235, −8.651680675321017936285273133281, −7.40947238012626396845926962101, −6.21101084864347639210962569250, −4.93151201178692053020941328368, −3.53742595638072786183789211796, −1.69518179825157615838052908524,
3.19431480777336951698709579511, 4.82483220358578569172754879461, 5.74020866811977916263342817450, 6.28818332010741903416619855056, 8.126174268687668573070148696108, 9.514145424467487276232994665894, 10.37067955291399168118209929665, 11.49938671630806230987172673337, 12.82587942352736248085253475693, 13.43666574230434387669881363155
