| L(s) = 1 | − 2-s + 0.670·3-s + 4-s + 1.45·5-s − 0.670·6-s + 0.581·7-s − 8-s − 2.55·9-s − 1.45·10-s + 11-s + 0.670·12-s − 1.74·13-s − 0.581·14-s + 0.973·15-s + 16-s + 0.878·17-s + 2.55·18-s − 1.44·19-s + 1.45·20-s + 0.390·21-s − 22-s + 1.79·23-s − 0.670·24-s − 2.89·25-s + 1.74·26-s − 3.72·27-s + 0.581·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.387·3-s + 0.5·4-s + 0.649·5-s − 0.273·6-s + 0.219·7-s − 0.353·8-s − 0.850·9-s − 0.459·10-s + 0.301·11-s + 0.193·12-s − 0.485·13-s − 0.155·14-s + 0.251·15-s + 0.250·16-s + 0.212·17-s + 0.601·18-s − 0.331·19-s + 0.324·20-s + 0.0851·21-s − 0.213·22-s + 0.374·23-s − 0.136·24-s − 0.578·25-s + 0.343·26-s − 0.716·27-s + 0.109·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.670T + 3T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 7 | \( 1 - 0.581T + 7T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.878T + 17T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + 4.72T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 - 9.47T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 + 5.86T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 + 8.26T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.020621234983923647043897801393, −7.54349696719737616033117254369, −6.59056633835590136992637559829, −5.87184170927623886387326239166, −5.26158437673713081989439259413, −4.10969520808147670910170886965, −3.10735680578958072966215479207, −2.30949902641556043652572826069, −1.50533095946239576829056062405, 0,
1.50533095946239576829056062405, 2.30949902641556043652572826069, 3.10735680578958072966215479207, 4.10969520808147670910170886965, 5.26158437673713081989439259413, 5.87184170927623886387326239166, 6.59056633835590136992637559829, 7.54349696719737616033117254369, 8.020621234983923647043897801393
