| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 2·13-s − 15-s + 16-s + 4·17-s − 18-s + 20-s + 8·23-s + 24-s + 25-s + 2·26-s − 27-s + 30-s − 2·31-s − 32-s − 4·34-s + 36-s + 8·37-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.182·30-s − 0.359·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 1.31·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.709580641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709580641\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.03309898763879, −12.75936845940198, −12.01432007492029, −11.83478134001424, −11.06243746353734, −10.91780475982662, −10.23765432165626, −9.864573284025061, −9.505342609384031, −8.831435227687869, −8.615933076882388, −7.664108165801840, −7.466795455629058, −6.967565180226378, −6.431057016560943, −5.666722771393693, −5.618290226876548, −4.870858056444430, −4.307369834080705, −3.627710744355534, −2.723277939563004, −2.632179166950287, −1.585968323637248, −1.110122961698531, −0.4955081855221097,
0.4955081855221097, 1.110122961698531, 1.585968323637248, 2.632179166950287, 2.723277939563004, 3.627710744355534, 4.307369834080705, 4.870858056444430, 5.618290226876548, 5.666722771393693, 6.431057016560943, 6.967565180226378, 7.466795455629058, 7.664108165801840, 8.615933076882388, 8.831435227687869, 9.505342609384031, 9.864573284025061, 10.23765432165626, 10.91780475982662, 11.06243746353734, 11.83478134001424, 12.01432007492029, 12.75936845940198, 13.03309898763879
