| L(s) = 1 | − 2-s + 4-s − 5-s + 3·7-s − 8-s + 10-s − 3·13-s − 3·14-s + 16-s + 17-s + 5·19-s − 20-s + 6·23-s + 25-s + 3·26-s + 3·28-s + 29-s − 32-s − 34-s − 3·35-s + 9·37-s − 5·38-s + 40-s + 4·41-s + 4·43-s − 6·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s − 0.832·13-s − 0.801·14-s + 1/4·16-s + 0.242·17-s + 1.14·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s + 0.588·26-s + 0.566·28-s + 0.185·29-s − 0.176·32-s − 0.171·34-s − 0.507·35-s + 1.47·37-s − 0.811·38-s + 0.158·40-s + 0.624·41-s + 0.609·43-s − 0.884·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677020118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677020118\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.70170208306621, −15.95644532421348, −15.44402407162509, −14.80252949095672, −14.42618612404510, −13.84092243379789, −12.87125540752551, −12.42868208704808, −11.64903660781651, −11.26833600001212, −10.87766012686319, −9.921628362584320, −9.542707423445634, −8.797705489120759, −8.165468743338872, −7.560849536214886, −7.293904465899393, −6.392786273615375, −5.468756963087759, −4.912765098409490, −4.234446385289888, −3.160625407725186, −2.521144018076010, −1.470753755413694, −0.7248933581968147,
0.7248933581968147, 1.470753755413694, 2.521144018076010, 3.160625407725186, 4.234446385289888, 4.912765098409490, 5.468756963087759, 6.392786273615375, 7.293904465899393, 7.560849536214886, 8.165468743338872, 8.797705489120759, 9.542707423445634, 9.921628362584320, 10.87766012686319, 11.26833600001212, 11.64903660781651, 12.42868208704808, 12.87125540752551, 13.84092243379789, 14.42618612404510, 14.80252949095672, 15.44402407162509, 15.95644532421348, 16.70170208306621
