| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 12-s − 13-s + 14-s − 15-s + 16-s + 3·17-s − 18-s − 19-s + 20-s + 21-s + 9·23-s + 24-s − 4·25-s + 26-s − 27-s − 28-s − 5·29-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 − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 1.87·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4770513180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4770513180\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.62873886856478, −12.98323184217916, −12.72200413058540, −12.33725408971372, −11.42094079275946, −11.31928103655923, −10.79453903702386, −10.13813358631138, −9.821182136820387, −9.253030192308041, −9.010345999135095, −8.160035339590763, −7.736458923031591, −7.078320067831699, −6.775857568377524, −6.188043092540628, −5.495840097366359, −5.275339426815301, −4.566332916446138, −3.620970866175280, −3.284720616435133, −2.487982589773222, −1.692974339733917, −1.311918949922947, −0.2532779115687856,
0.2532779115687856, 1.311918949922947, 1.692974339733917, 2.487982589773222, 3.284720616435133, 3.620970866175280, 4.566332916446138, 5.275339426815301, 5.495840097366359, 6.188043092540628, 6.775857568377524, 7.078320067831699, 7.736458923031591, 8.160035339590763, 9.010345999135095, 9.253030192308041, 9.821182136820387, 10.13813358631138, 10.79453903702386, 11.31928103655923, 11.42094079275946, 12.33725408971372, 12.72200413058540, 12.98323184217916, 13.62873886856478
