L(s) = 1 | + (−2.90 − 1.67i)5-s − 3.54·7-s − 0.251i·11-s + (−4.29 + 2.47i)13-s + (3.11 + 1.80i)17-s + (−4.15 + 1.30i)19-s + (−3.89 + 2.24i)23-s + (3.11 + 5.39i)25-s + (2.27 + 3.94i)29-s − 5.28i·31-s + (10.2 + 5.94i)35-s − 6.47i·37-s + (−4.96 + 8.59i)41-s + (3.38 − 5.86i)43-s + (8.96 − 5.17i)47-s + ⋯ |
L(s) = 1 | + (−1.29 − 0.749i)5-s − 1.34·7-s − 0.0758i·11-s + (−1.18 + 0.687i)13-s + (0.756 + 0.436i)17-s + (−0.954 + 0.298i)19-s + (−0.811 + 0.468i)23-s + (0.622 + 1.07i)25-s + (0.423 + 0.732i)29-s − 0.949i·31-s + (1.73 + 1.00i)35-s − 1.06i·37-s + (−0.774 + 1.34i)41-s + (0.516 − 0.894i)43-s + (1.30 − 0.754i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5906653211\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5906653211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4.15 - 1.30i)T \) |
good | 5 | \( 1 + (2.90 + 1.67i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 0.251iT - 11T^{2} \) |
| 13 | \( 1 + (4.29 - 2.47i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.11 - 1.80i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.89 - 2.24i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.27 - 3.94i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.28iT - 31T^{2} \) |
| 37 | \( 1 + 6.47iT - 37T^{2} \) |
| 41 | \( 1 + (4.96 - 8.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.38 + 5.86i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.96 + 5.17i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.217 + 0.377i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.06 - 3.56i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.46 + 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.273 - 0.157i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.10 + 7.11i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.356 + 0.616i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.57 - 4.37i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.31iT - 83T^{2} \) |
| 89 | \( 1 + (1.20 + 2.09i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.36 - 5.40i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.808082702995476710545546561837, −7.977440292383154394113248959188, −7.40279544220019526378551829839, −6.57748937402136397219579238790, −5.74948562257889362629516976764, −4.71350925833497808630684130651, −3.96649587276635932425315718814, −3.35607094912026189191110639183, −2.09147314444021357922467573050, −0.45712187904732907422998685591,
0.44386397145313041302127477723, 2.56634140393665756826382843705, 3.11014546812029585098445296899, 3.95179784279931303163140566051, 4.79501033557318891112481926291, 5.95850428693568675024433294121, 6.69942386030773732294692590575, 7.34457636835506351244555024816, 7.914662146137865393225471831773, 8.783344429032772627027853994913