L(s) = 1 | − 0.544·3-s + 0.962·5-s + 3.25·7-s − 2.70·9-s + 0.883·11-s − 4.48·13-s − 0.523·15-s − 3.18·17-s − 4.12·19-s − 1.77·21-s − 3.73·23-s − 4.07·25-s + 3.10·27-s + 6.33·29-s + 8.47·31-s − 0.480·33-s + 3.13·35-s + 11.0·37-s + 2.43·39-s + 1.41·41-s + 9.31·43-s − 2.60·45-s + 5.82·47-s + 3.58·49-s + 1.73·51-s − 7.06·53-s + 0.850·55-s + ⋯ |
L(s) = 1 | − 0.314·3-s + 0.430·5-s + 1.22·7-s − 0.901·9-s + 0.266·11-s − 1.24·13-s − 0.135·15-s − 0.771·17-s − 0.946·19-s − 0.386·21-s − 0.779·23-s − 0.814·25-s + 0.597·27-s + 1.17·29-s + 1.52·31-s − 0.0836·33-s + 0.529·35-s + 1.81·37-s + 0.390·39-s + 0.220·41-s + 1.41·43-s − 0.388·45-s + 0.849·47-s + 0.511·49-s + 0.242·51-s − 0.970·53-s + 0.114·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
good | 3 | \( 1 + 0.544T + 3T^{2} \) |
| 5 | \( 1 - 0.962T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 0.883T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 4.12T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 9.31T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 + 7.06T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 9.15T + 67T^{2} \) |
| 71 | \( 1 - 4.57T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 7.20T + 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 - 0.838T + 89T^{2} \) |
| 97 | \( 1 - 0.475T + 97T^{2} \) |
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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
−7.63637052640872530801010778203, −6.47028897647393562458970710527, −6.13787390305214112790801999054, −5.33906718597048277065143609382, −4.49386020815810041381238353436, −4.30279468198870210611714088132, −2.64579118216129940511804190376, −2.39479981962883720170066297438, −1.26319367492338082619405631198, 0,
1.26319367492338082619405631198, 2.39479981962883720170066297438, 2.64579118216129940511804190376, 4.30279468198870210611714088132, 4.49386020815810041381238353436, 5.33906718597048277065143609382, 6.13787390305214112790801999054, 6.47028897647393562458970710527, 7.63637052640872530801010778203