| L(s) = 1 | − 2-s + 2·3-s − 4-s − 5-s − 2·6-s + 4·7-s + 3·8-s + 9-s + 10-s + 4·11-s − 2·12-s − 4·14-s − 2·15-s − 16-s − 8·17-s − 18-s + 4·19-s + 20-s + 8·21-s − 4·22-s + 2·23-s + 6·24-s + 25-s − 4·27-s − 4·28-s − 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.577·12-s − 1.06·14-s − 0.516·15-s − 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 1.74·21-s − 0.852·22-s + 0.417·23-s + 1.22·24-s + 1/5·25-s − 0.769·27-s − 0.755·28-s − 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057069688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057069688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.29030496383911904269281815554, −11.67404196451839618122260474934, −10.95042563470151669189144154002, −9.406906855544247656461754170978, −8.774417290846673446056763114030, −8.125055688796012797619570937515, −7.12181256175802188916168498633, −4.89232440280039987399206936754, −3.83108286299930817257651347819, −1.76743844147568883236115154570,
1.76743844147568883236115154570, 3.83108286299930817257651347819, 4.89232440280039987399206936754, 7.12181256175802188916168498633, 8.125055688796012797619570937515, 8.774417290846673446056763114030, 9.406906855544247656461754170978, 10.95042563470151669189144154002, 11.67404196451839618122260474934, 13.29030496383911904269281815554
