| L(s) = 1 | + (−1.15 − 0.819i)2-s + (1.33 − 1.10i)3-s + (0.656 + 1.88i)4-s + (0.128 + 0.378i)5-s + (−2.44 + 0.179i)6-s + (−1.38 − 1.80i)7-s + (0.791 − 2.71i)8-s + (0.560 − 2.94i)9-s + (0.162 − 0.542i)10-s + (−5.67 + 0.371i)11-s + (2.96 + 1.79i)12-s + (−1.49 − 1.70i)13-s + (0.116 + 3.21i)14-s + (0.590 + 0.363i)15-s + (−3.13 + 2.48i)16-s + (0.935 − 0.935i)17-s + ⋯ |
| L(s) = 1 | + (−0.814 − 0.579i)2-s + (0.770 − 0.637i)3-s + (0.328 + 0.944i)4-s + (0.0575 + 0.169i)5-s + (−0.997 + 0.0731i)6-s + (−0.524 − 0.683i)7-s + (0.279 − 0.960i)8-s + (0.186 − 0.982i)9-s + (0.0513 − 0.171i)10-s + (−1.71 + 0.112i)11-s + (0.855 + 0.518i)12-s + (−0.414 − 0.472i)13-s + (0.0312 + 0.860i)14-s + (0.152 + 0.0938i)15-s + (−0.784 + 0.620i)16-s + (0.227 − 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0820516 - 0.759112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0820516 - 0.759112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.15 + 0.819i)T \) |
| 3 | \( 1 + (-1.33 + 1.10i)T \) |
| good | 5 | \( 1 + (-0.128 - 0.378i)T + (-3.96 + 3.04i)T^{2} \) |
| 7 | \( 1 + (1.38 + 1.80i)T + (-1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (5.67 - 0.371i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (1.49 + 1.70i)T + (-1.69 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.935 + 0.935i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.719 - 3.61i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (4.47 + 3.43i)T + (5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (3.89 + 7.89i)T + (-17.6 + 23.0i)T^{2} \) |
| 31 | \( 1 + (-3.85 + 2.22i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.930 + 4.67i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (5.12 + 3.93i)T + (10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (-0.767 - 11.7i)T + (-42.6 + 5.61i)T^{2} \) |
| 47 | \( 1 + (-1.78 - 6.67i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.873 - 1.30i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.25 - 3.70i)T + (-46.8 + 35.9i)T^{2} \) |
| 61 | \( 1 + (-3.00 + 1.48i)T + (37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (-0.404 + 6.16i)T + (-66.4 - 8.74i)T^{2} \) |
| 71 | \( 1 + (-1.44 + 3.48i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.52 - 13.3i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (11.6 - 3.11i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.52 - 2.55i)T + (65.8 + 50.5i)T^{2} \) |
| 89 | \( 1 + (2.28 + 0.947i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-8.69 - 5.01i)T + (48.5 + 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.993621202889750647401436329906, −9.794473800711592673430947390630, −8.311985187792699539532417002516, −7.86942830527286485119642435434, −7.15835041968655297983915990975, −6.02650355963319582816488631483, −4.21240453513638046162961501702, −3.01387944059369023202683019719, −2.27521747927144876326845260313, −0.47567437115031629320774796549,
2.11773354302224468399882435275, 3.16151318835687577361550443933, 4.97587322993168103774795963461, 5.47682995101148789823663670675, 6.91791439028409579318834173815, 7.77349379061973168971159694617, 8.640503443046645459236624338421, 9.215031460163719382817215872508, 10.10792689373401723343231324289, 10.62237840196023687405654136481
