| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 14-s + 15-s + 16-s + 4·17-s − 18-s − 6·19-s + 20-s − 21-s − 24-s + 25-s + 27-s − 28-s − 30-s − 8·31-s − 32-s − 4·34-s − 35-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 + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.364855929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.364855929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.81039825525517, −12.95550016629970, −12.78568262861966, −12.25368999575572, −11.74973344117859, −10.81793914399951, −10.71042654221854, −10.28127887912855, −9.423614019870742, −9.281414656913413, −8.941661355854786, −8.133004052659814, −7.766067390620238, −7.325509812089958, −6.674599425186538, −6.143525754958098, −5.705989155485909, −5.086333127701852, −4.124940595742433, −3.876934235962948, −3.028164580958770, −2.451699019591759, −2.045773062584972, −1.204572586598496, −0.5414931183499441,
0.5414931183499441, 1.204572586598496, 2.045773062584972, 2.451699019591759, 3.028164580958770, 3.876934235962948, 4.124940595742433, 5.086333127701852, 5.705989155485909, 6.143525754958098, 6.674599425186538, 7.325509812089958, 7.766067390620238, 8.133004052659814, 8.941661355854786, 9.281414656913413, 9.423614019870742, 10.28127887912855, 10.71042654221854, 10.81793914399951, 11.74973344117859, 12.25368999575572, 12.78568262861966, 12.95550016629970, 13.81039825525517
