| L(s) = 1 | − 4.21·2-s − 3·3-s + 9.74·4-s + 8.49·5-s + 12.6·6-s − 7.33·8-s + 9·9-s − 35.7·10-s − 52.9·11-s − 29.2·12-s + 13·13-s − 25.4·15-s − 47.0·16-s − 0.773·17-s − 37.9·18-s − 18.8·19-s + 82.7·20-s + 222.·22-s + 80.1·23-s + 21.9·24-s − 52.7·25-s − 54.7·26-s − 27·27-s + 241.·29-s + 107.·30-s + 159.·31-s + 256.·32-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.577·3-s + 1.21·4-s + 0.760·5-s + 0.859·6-s − 0.324·8-s + 0.333·9-s − 1.13·10-s − 1.45·11-s − 0.702·12-s + 0.277·13-s − 0.438·15-s − 0.735·16-s − 0.0110·17-s − 0.496·18-s − 0.228·19-s + 0.925·20-s + 2.16·22-s + 0.726·23-s + 0.187·24-s − 0.422·25-s − 0.413·26-s − 0.192·27-s + 1.54·29-s + 0.653·30-s + 0.921·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 4.21T + 8T^{2} \) |
| 5 | \( 1 - 8.49T + 125T^{2} \) |
| 11 | \( 1 + 52.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 0.773T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 256.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 162.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 559.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 635.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 70.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 334.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 416.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 400.T + 9.12e5T^{2} \) |
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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
−8.260056033911451018106553030708, −8.102082210138254713318758990178, −6.89095833749109136278715896884, −6.38083436987597964543907569232, −5.31512921074588512331950474348, −4.62974694065368396936737900707, −2.95952546767129860689441801785, −2.01982395544726912580253611653, −1.01133478639080669019514682793, 0,
1.01133478639080669019514682793, 2.01982395544726912580253611653, 2.95952546767129860689441801785, 4.62974694065368396936737900707, 5.31512921074588512331950474348, 6.38083436987597964543907569232, 6.89095833749109136278715896884, 8.102082210138254713318758990178, 8.260056033911451018106553030708
