| L(s) = 1 | − 3-s + 9-s + 1.50·11-s − 13-s + 2.72·17-s + 0.726·19-s + 4.72·23-s − 27-s − 7.55·29-s − 3.00·31-s − 1.50·33-s + 5.00·37-s + 39-s + 5.78·41-s − 2.72·43-s + 10.2·47-s − 7·49-s − 2.72·51-s + 7.55·53-s − 0.726·57-s − 12.5·59-s + 6.28·61-s + 12.5·67-s − 4.72·69-s + 4.77·71-s − 12.0·73-s + 5.27·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.333·9-s + 0.453·11-s − 0.277·13-s + 0.661·17-s + 0.166·19-s + 0.985·23-s − 0.192·27-s − 1.40·29-s − 0.540·31-s − 0.261·33-s + 0.823·37-s + 0.160·39-s + 0.903·41-s − 0.415·43-s + 1.49·47-s − 49-s − 0.381·51-s + 1.03·53-s − 0.0962·57-s − 1.62·59-s + 0.804·61-s + 1.53·67-s − 0.569·69-s + 0.567·71-s − 1.40·73-s + 0.593·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633597789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633597789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 - 0.726T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 5.00T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 + 2.72T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.63285084897664292006242272242, −7.22710974206921604680491590414, −6.44623029456296488930362747074, −5.68684936464556459182917286618, −5.20970765512806437204505101054, −4.30769739553919629288244953081, −3.62932430305048909538900389536, −2.69096946525478149690527944364, −1.63348048511831322267628921620, −0.67998279653304173848770218470,
0.67998279653304173848770218470, 1.63348048511831322267628921620, 2.69096946525478149690527944364, 3.62932430305048909538900389536, 4.30769739553919629288244953081, 5.20970765512806437204505101054, 5.68684936464556459182917286618, 6.44623029456296488930362747074, 7.22710974206921604680491590414, 7.63285084897664292006242272242
