L(s) = 1 | + (−0.207 + 0.358i)2-s + (0.914 + 1.58i)4-s + (−0.5 + 0.866i)5-s + 1.82·7-s − 1.58·8-s + (−0.207 − 0.358i)10-s + 2.82·11-s + (0.914 + 1.58i)13-s + (−0.378 + 0.655i)14-s + (−1.49 + 2.59i)16-s + (−0.585 + 1.01i)17-s + (4 − 1.73i)19-s − 1.82·20-s + (−0.585 + 1.01i)22-s + (−0.414 − 0.717i)23-s + ⋯ |
L(s) = 1 | + (−0.146 + 0.253i)2-s + (0.457 + 0.791i)4-s + (−0.223 + 0.387i)5-s + 0.691·7-s − 0.560·8-s + (−0.0654 − 0.113i)10-s + 0.852·11-s + (0.253 + 0.439i)13-s + (−0.101 + 0.175i)14-s + (−0.374 + 0.649i)16-s + (−0.142 + 0.246i)17-s + (0.917 − 0.397i)19-s − 0.408·20-s + (−0.124 + 0.216i)22-s + (−0.0863 − 0.149i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11763 + 1.23277i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11763 + 1.23277i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 2 | \( 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + (-0.914 - 1.58i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.585 - 1.01i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.414 + 0.717i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.82 - 8.36i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 + (-1.41 + 2.44i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.91 - 6.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.58 + 2.74i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.15 - 7.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.74 + 4.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5 + 8.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.74 - 8.21i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.67 + 2.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + (-6.24 - 10.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.59086765557681449112254109521, −9.307887442883313096730002718773, −8.639325829248419407696360072546, −7.81120901884500628590531389722, −6.98640162379484144943588917881, −6.40437767461734359444151609195, −5.07077938489908334972766734758, −3.92471164930328643949208812155, −3.05356233117805522530622085727, −1.65939686926094256625508042628,
0.918218696833690515175329225242, 1.99465804068466328322066550493, 3.43197614292778025536600580339, 4.66718936476540772382927057986, 5.55215288811809189456164127551, 6.40476831805804803909153137550, 7.44159217249037496392029981502, 8.326299055699953988930225555920, 9.258567081997665989987758629663, 9.947722280057485080315481949949