| L(s) = 1 | + (−0.994 − 0.104i)2-s + (−0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (0.207 − 0.978i)5-s + (0.207 + 0.978i)6-s + (0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (−0.169 + 1.60i)11-s + (−0.104 − 0.994i)12-s + (−0.614 + 0.0646i)13-s + (−0.913 + 0.406i)14-s + (−0.994 + 0.104i)15-s + (0.913 + 0.406i)16-s + (0.309 − 0.951i)17-s + ⋯ |
| L(s) = 1 | + (−0.994 − 0.104i)2-s + (−0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (0.207 − 0.978i)5-s + (0.207 + 0.978i)6-s + (0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (−0.169 + 1.60i)11-s + (−0.104 − 0.994i)12-s + (−0.614 + 0.0646i)13-s + (−0.913 + 0.406i)14-s + (−0.994 + 0.104i)15-s + (0.913 + 0.406i)16-s + (0.309 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6204532170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6204532170\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.994 + 0.104i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.207 + 0.978i)T \) |
| good | 7 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.169 - 1.60i)T + (-0.978 - 0.207i)T^{2} \) |
| 13 | \( 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \) |
| 29 | \( 1 + (1.20 + 1.08i)T + (0.104 + 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.743 + 0.669i)T + (0.104 - 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.0646 + 0.614i)T + (-0.978 + 0.207i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \) |
| 53 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 61 | \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 67 | \( 1 + (0.413 + 0.459i)T + (-0.104 + 0.994i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.104 - 0.994i)T^{2} \) |
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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
−9.226472371847414081449275210215, −8.205863051506280667746230349665, −7.52759786822485135997806813516, −7.25696393330014403234721845149, −6.15216425842776780214445623537, −5.10179408481285663456063775452, −4.43487228863497813981026443978, −2.46342427029628528858577863982, −1.86347575517098944626187937531, −0.66040458067767676132024865087,
1.68113029097783574333637093811, 3.00986057809042753475810141481, 3.61979358772136611609535082611, 5.30748131074420226167798811409, 5.76688756705578511127830545647, 6.49688673688781678817494558155, 7.72884430994956463618408175981, 8.239780246405279260624480061220, 9.056392418723939927059573233960, 9.820031312636638618846781179965
