| L(s) = 1 | + 1.97·2-s − 3.27·3-s + 1.88·4-s + 5-s − 6.45·6-s − 0.220·8-s + 7.71·9-s + 1.97·10-s + 2.33·11-s − 6.17·12-s + 0.629·13-s − 3.27·15-s − 4.21·16-s + 2.94·17-s + 15.2·18-s − 5.49·19-s + 1.88·20-s + 4.59·22-s + 0.988·23-s + 0.721·24-s + 25-s + 1.24·26-s − 15.4·27-s − 0.971·29-s − 6.45·30-s − 5.16·31-s − 7.86·32-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 1.88·3-s + 0.944·4-s + 0.447·5-s − 2.63·6-s − 0.0779·8-s + 2.57·9-s + 0.623·10-s + 0.703·11-s − 1.78·12-s + 0.174·13-s − 0.844·15-s − 1.05·16-s + 0.714·17-s + 3.58·18-s − 1.25·19-s + 0.422·20-s + 0.980·22-s + 0.206·23-s + 0.147·24-s + 0.200·25-s + 0.243·26-s − 2.96·27-s − 0.180·29-s − 1.17·30-s − 0.927·31-s − 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 - 0.629T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 23 | \( 1 - 0.988T + 23T^{2} \) |
| 29 | \( 1 + 0.971T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 41 | \( 1 - 0.554T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 5.29T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 - 5.44T + 73T^{2} \) |
| 79 | \( 1 - 5.57T + 79T^{2} \) |
| 83 | \( 1 - 3.60T + 83T^{2} \) |
| 89 | \( 1 - 9.82T + 89T^{2} \) |
| 97 | \( 1 - 7.92T + 97T^{2} \) |
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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
−6.84682250609884407469482561045, −6.32423425677077587770934163211, −5.99783366308233699879389874534, −5.25339101617698056922485446761, −4.83775064334176320436910970013, −4.07738318305422519675426980002, −3.50005172530550469435207623137, −2.16850355568253644366302915791, −1.25759920776434588475300454044, 0,
1.25759920776434588475300454044, 2.16850355568253644366302915791, 3.50005172530550469435207623137, 4.07738318305422519675426980002, 4.83775064334176320436910970013, 5.25339101617698056922485446761, 5.99783366308233699879389874534, 6.32423425677077587770934163211, 6.84682250609884407469482561045
