| L(s) = 1 | + 0.823·2-s − 0.960·3-s − 1.32·4-s + 2.98·5-s − 0.791·6-s − 7-s − 2.73·8-s − 2.07·9-s + 2.45·10-s + 1.26·12-s − 2.20·13-s − 0.823·14-s − 2.86·15-s + 0.390·16-s − 4.40·17-s − 1.71·18-s + 1.72·19-s − 3.94·20-s + 0.960·21-s − 8.39·23-s + 2.62·24-s + 3.91·25-s − 1.81·26-s + 4.87·27-s + 1.32·28-s − 3.29·29-s − 2.36·30-s + ⋯ |
| L(s) = 1 | + 0.582·2-s − 0.554·3-s − 0.660·4-s + 1.33·5-s − 0.322·6-s − 0.377·7-s − 0.967·8-s − 0.692·9-s + 0.777·10-s + 0.366·12-s − 0.610·13-s − 0.220·14-s − 0.740·15-s + 0.0976·16-s − 1.06·17-s − 0.403·18-s + 0.395·19-s − 0.882·20-s + 0.209·21-s − 1.75·23-s + 0.536·24-s + 0.782·25-s − 0.355·26-s + 0.938·27-s + 0.249·28-s − 0.611·29-s − 0.431·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.823T + 2T^{2} \) |
| 3 | \( 1 + 0.960T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 13 | \( 1 + 2.20T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 + 8.39T + 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 + 5.39T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 - 9.39T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 + 7.63T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−9.896664492318080218068868247290, −9.043313033510566361232860348848, −8.283740362226492597565048252640, −6.72028297305541500841957948675, −6.04679643375846784297203083097, −5.39191367522529546264150136695, −4.60946958450508369006429540874, −3.28453108231070537346553490791, −2.10000486205875759830003682145, 0,
2.10000486205875759830003682145, 3.28453108231070537346553490791, 4.60946958450508369006429540874, 5.39191367522529546264150136695, 6.04679643375846784297203083097, 6.72028297305541500841957948675, 8.283740362226492597565048252640, 9.043313033510566361232860348848, 9.896664492318080218068868247290
