| L(s) = 1 | + 0.173·2-s + 1.11·3-s − 1.97·4-s + 5-s + 0.193·6-s − 7-s − 0.687·8-s − 1.75·9-s + 0.173·10-s − 5.19·11-s − 2.20·12-s + 3.30·13-s − 0.173·14-s + 1.11·15-s + 3.82·16-s − 5.71·17-s − 0.303·18-s − 1.66·19-s − 1.97·20-s − 1.11·21-s − 0.899·22-s + 23-s − 0.768·24-s + 25-s + 0.572·26-s − 5.30·27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | + 0.122·2-s + 0.645·3-s − 0.985·4-s + 0.447·5-s + 0.0790·6-s − 0.377·7-s − 0.243·8-s − 0.583·9-s + 0.0547·10-s − 1.56·11-s − 0.635·12-s + 0.917·13-s − 0.0462·14-s + 0.288·15-s + 0.955·16-s − 1.38·17-s − 0.0714·18-s − 0.382·19-s − 0.440·20-s − 0.243·21-s − 0.191·22-s + 0.208·23-s − 0.156·24-s + 0.200·25-s + 0.112·26-s − 1.02·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.173T + 2T^{2} \) |
| 3 | \( 1 - 1.11T + 3T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 29 | \( 1 + 9.33T + 29T^{2} \) |
| 31 | \( 1 + 9.77T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.98T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 + 5.79T + 61T^{2} \) |
| 67 | \( 1 - 0.111T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 5.56T + 89T^{2} \) |
| 97 | \( 1 + 0.969T + 97T^{2} \) |
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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
−9.501969869977991538345955328901, −9.071457969106550601367046961937, −8.299213172691908633517175738295, −7.49791204651567986206358789251, −6.04115342867661500091812385148, −5.45022522361365321865928326722, −4.27800235196084876090789714861, −3.25999311310528020476628193483, −2.20957952254265912530327853006, 0,
2.20957952254265912530327853006, 3.25999311310528020476628193483, 4.27800235196084876090789714861, 5.45022522361365321865928326722, 6.04115342867661500091812385148, 7.49791204651567986206358789251, 8.299213172691908633517175738295, 9.071457969106550601367046961937, 9.501969869977991538345955328901
