| L(s) = 1 | + 1.14·3-s − 3.94·7-s − 1.67·9-s + 2.65·11-s − 3.06·13-s − 0.579·17-s − 6.95·19-s − 4.53·21-s + 1.18·23-s − 5.37·27-s + 7.49·29-s − 6.65·31-s + 3.05·33-s − 11.4·37-s − 3.51·39-s + 1.86·41-s − 0.851·43-s − 6.56·47-s + 8.57·49-s − 0.666·51-s − 10.4·53-s − 7.99·57-s + 0.184·59-s + 8.97·61-s + 6.62·63-s + 5.64·67-s + 1.35·69-s + ⋯ |
| L(s) = 1 | + 0.663·3-s − 1.49·7-s − 0.559·9-s + 0.800·11-s − 0.849·13-s − 0.140·17-s − 1.59·19-s − 0.989·21-s + 0.246·23-s − 1.03·27-s + 1.39·29-s − 1.19·31-s + 0.531·33-s − 1.87·37-s − 0.563·39-s + 0.291·41-s − 0.129·43-s − 0.958·47-s + 1.22·49-s − 0.0932·51-s − 1.43·53-s − 1.05·57-s + 0.0239·59-s + 1.14·61-s + 0.835·63-s + 0.689·67-s + 0.163·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 + 0.579T + 17T^{2} \) |
| 19 | \( 1 + 6.95T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 + 0.851T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 0.184T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 67 | \( 1 - 5.64T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 - 0.304T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.419084202508067488919171088039, −8.872766248533614873009224079369, −8.054949922540494814184982099006, −6.77391279590002041057024135848, −6.46369507730296804351798216968, −5.20452605001882168729986532361, −3.92206660313707795004923820637, −3.14735217346602350076682139128, −2.15156421233656875992930836958, 0,
2.15156421233656875992930836958, 3.14735217346602350076682139128, 3.92206660313707795004923820637, 5.20452605001882168729986532361, 6.46369507730296804351798216968, 6.77391279590002041057024135848, 8.054949922540494814184982099006, 8.872766248533614873009224079369, 9.419084202508067488919171088039
