| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s − 5·13-s − 14-s + 15-s + 16-s + 18-s − 20-s + 21-s + 6·22-s + 6·23-s − 24-s + 25-s − 5·26-s − 27-s − 28-s − 6·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.218·21-s + 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.75664994547372, −16.43554566958720, −15.47410988113275, −15.01421014025037, −14.69728665772738, −13.93327637858641, −13.40681863898472, −12.47841671109537, −12.30120809435260, −11.79077989824837, −11.15716220442066, −10.62028238710020, −9.813225325754132, −9.201168793804266, −8.691832002955382, −7.481490571648136, −7.067362620102841, −6.700124811829989, −5.807973100410090, −5.185225853846472, −4.536345117409298, −3.779593761926864, −3.291924525593791, −2.155371750626999, −1.256911846465467, 0,
1.256911846465467, 2.155371750626999, 3.291924525593791, 3.779593761926864, 4.536345117409298, 5.185225853846472, 5.807973100410090, 6.700124811829989, 7.067362620102841, 7.481490571648136, 8.691832002955382, 9.201168793804266, 9.813225325754132, 10.62028238710020, 11.15716220442066, 11.79077989824837, 12.30120809435260, 12.47841671109537, 13.40681863898472, 13.93327637858641, 14.69728665772738, 15.01421014025037, 15.47410988113275, 16.43554566958720, 16.75664994547372
