| L(s) = 1 | − 0.673·2-s − 0.811·3-s − 1.54·4-s + 4.31·5-s + 0.546·6-s + 2.38·8-s − 2.34·9-s − 2.90·10-s − 5.15·11-s + 1.25·12-s − 3.34·13-s − 3.49·15-s + 1.48·16-s + 17-s + 1.57·18-s − 4.18·19-s − 6.66·20-s + 3.46·22-s + 2.49·23-s − 1.93·24-s + 13.5·25-s + 2.25·26-s + 4.33·27-s − 5.52·29-s + 2.35·30-s − 4.84·31-s − 5.77·32-s + ⋯ |
| L(s) = 1 | − 0.476·2-s − 0.468·3-s − 0.773·4-s + 1.92·5-s + 0.223·6-s + 0.844·8-s − 0.780·9-s − 0.918·10-s − 1.55·11-s + 0.362·12-s − 0.928·13-s − 0.903·15-s + 0.371·16-s + 0.242·17-s + 0.371·18-s − 0.959·19-s − 1.49·20-s + 0.739·22-s + 0.521·23-s − 0.395·24-s + 2.71·25-s + 0.441·26-s + 0.834·27-s − 1.02·29-s + 0.430·30-s − 0.869·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.673T + 2T^{2} \) |
| 3 | \( 1 + 0.811T + 3T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 4.84T + 31T^{2} \) |
| 37 | \( 1 + 0.193T + 37T^{2} \) |
| 41 | \( 1 + 6.08T + 41T^{2} \) |
| 43 | \( 1 - 1.00T + 43T^{2} \) |
| 47 | \( 1 + 7.78T + 47T^{2} \) |
| 53 | \( 1 - 0.341T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 6.08T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 + 6.02T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−9.817265749320002768053065957523, −9.103443050796951918811635430252, −8.329261249928122845293906663186, −7.24492754113958857708551783726, −6.04310223357292462212045780186, −5.32451187166226081086407392825, −4.87668264076304642628147913124, −2.90568865727713229724956303645, −1.84709664238402488195220135699, 0,
1.84709664238402488195220135699, 2.90568865727713229724956303645, 4.87668264076304642628147913124, 5.32451187166226081086407392825, 6.04310223357292462212045780186, 7.24492754113958857708551783726, 8.329261249928122845293906663186, 9.103443050796951918811635430252, 9.817265749320002768053065957523
