L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s + (−2 − 3.46i)5-s + (1 − 1.73i)7-s − 3·8-s + 3.99·10-s + (−0.5 + 0.866i)11-s + (1 + 1.73i)13-s + (0.999 + 1.73i)14-s + (0.500 − 0.866i)16-s − 2·17-s − 6·19-s + (1.99 − 3.46i)20-s + (−0.499 − 0.866i)22-s + (2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.894 − 1.54i)5-s + (0.377 − 0.654i)7-s − 1.06·8-s + 1.26·10-s + (−0.150 + 0.261i)11-s + (0.277 + 0.480i)13-s + (0.267 + 0.462i)14-s + (0.125 − 0.216i)16-s − 0.485·17-s − 1.37·19-s + (0.447 − 0.774i)20-s + (−0.106 − 0.184i)22-s + (0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.207948177968772448133856251722, −8.832611423657132076547209976130, −8.017545465213814264107249330539, −7.43614320171919205216045652117, −6.54760725323309604010972060240, −5.26459297329502689742601361636, −4.36332847499736639683711378308, −3.59653643658300951519695607133, −1.70521221475953509159235261326, 0,
2.08796912368310969291413397023, 2.88093968080214712478354047342, 3.87005922360318111365880062926, 5.33660393000965542334658859514, 6.42754529641153022576734659541, 6.90755415708715459241708014316, 8.198791041195620481261305819974, 8.726524999166781399492757690371, 10.06008018455552287803754459286