L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 3.41·11-s − 12-s + 2·13-s − 14-s − 15-s + 16-s − 3.26·17-s − 18-s − 3.41·19-s + 20-s − 21-s − 3.41·22-s + 23-s + 24-s + 25-s − 2·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.03·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.791·17-s − 0.235·18-s − 0.784·19-s + 0.223·20-s − 0.218·21-s − 0.728·22-s + 0.208·23-s + 0.204·24-s + 0.200·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.397775816\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397775816\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 9.57T + 67T^{2} \) |
| 71 | \( 1 + 9.57T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 + 0.156T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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
−8.521193596044076389741312578918, −7.50122139305648892105078515590, −6.75775601485698006089589886019, −6.25729836144522353770867097716, −5.57465920257156988028942161781, −4.53979234923654555512357177624, −3.87679803790337598087955834275, −2.57959559208125832895087879224, −1.68230379597080096388294680186, −0.78292712091054739650768518648,
0.78292712091054739650768518648, 1.68230379597080096388294680186, 2.57959559208125832895087879224, 3.87679803790337598087955834275, 4.53979234923654555512357177624, 5.57465920257156988028942161781, 6.25729836144522353770867097716, 6.75775601485698006089589886019, 7.50122139305648892105078515590, 8.521193596044076389741312578918