| L(s) = 1 | − 3.83·2-s + 6.69·4-s + 5·5-s + 4.98·8-s − 19.1·10-s + 6.91·11-s − 50.6·13-s − 72.7·16-s − 47.2·17-s + 39.0·19-s + 33.4·20-s − 26.4·22-s − 10.2·23-s + 25·25-s + 194.·26-s + 208.·29-s + 118.·31-s + 238.·32-s + 181.·34-s + 124.·37-s − 149.·38-s + 24.9·40-s − 224.·41-s + 211.·43-s + 46.2·44-s + 39.1·46-s − 497.·47-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.837·4-s + 0.447·5-s + 0.220·8-s − 0.606·10-s + 0.189·11-s − 1.08·13-s − 1.13·16-s − 0.674·17-s + 0.471·19-s + 0.374·20-s − 0.256·22-s − 0.0926·23-s + 0.200·25-s + 1.46·26-s + 1.33·29-s + 0.689·31-s + 1.31·32-s + 0.913·34-s + 0.553·37-s − 0.639·38-s + 0.0986·40-s − 0.854·41-s + 0.751·43-s + 0.158·44-s + 0.125·46-s − 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.83T + 8T^{2} \) |
| 11 | \( 1 - 6.91T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 10.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 224.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 383.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 82.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 304.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 256.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 415.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 508.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 49.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 227.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
| 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
−8.425870329944823556814823196217, −7.73431170325312338206155301872, −6.92387411622033518388242550443, −6.30867545697444816156673636441, −5.05967944494317566753845523734, −4.41708520588463059765950185810, −2.93662979084154236575245522380, −2.05741278803234579823892869302, −1.06037730096084578455300140298, 0,
1.06037730096084578455300140298, 2.05741278803234579823892869302, 2.93662979084154236575245522380, 4.41708520588463059765950185810, 5.05967944494317566753845523734, 6.30867545697444816156673636441, 6.92387411622033518388242550443, 7.73431170325312338206155301872, 8.425870329944823556814823196217
