| L(s) = 1 | − 0.501·2-s − 3·3-s − 7.74·4-s + 1.50·6-s − 17.4·7-s + 7.89·8-s + 9·9-s − 20.0·11-s + 23.2·12-s − 34.7·13-s + 8.74·14-s + 58.0·16-s + 88.8·17-s − 4.50·18-s − 103.·19-s + 52.3·21-s + 10.0·22-s − 54.9·23-s − 23.6·24-s + 17.4·26-s − 27·27-s + 135.·28-s − 29·29-s − 52.6·31-s − 92.2·32-s + 60.1·33-s − 44.5·34-s + ⋯ |
| L(s) = 1 | − 0.177·2-s − 0.577·3-s − 0.968·4-s + 0.102·6-s − 0.942·7-s + 0.348·8-s + 0.333·9-s − 0.549·11-s + 0.559·12-s − 0.742·13-s + 0.166·14-s + 0.906·16-s + 1.26·17-s − 0.0590·18-s − 1.24·19-s + 0.544·21-s + 0.0973·22-s − 0.498·23-s − 0.201·24-s + 0.131·26-s − 0.192·27-s + 0.912·28-s − 0.185·29-s − 0.304·31-s − 0.509·32-s + 0.317·33-s − 0.224·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1213826111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1213826111\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 0.501T + 8T^{2} \) |
| 7 | \( 1 + 17.4T + 343T^{2} \) |
| 11 | \( 1 + 20.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 54.9T + 1.21e4T^{2} \) |
| 31 | \( 1 + 52.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 194.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 85.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 494.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 70.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 622.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 215.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 926.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 77.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 324.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 504.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
| 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
−8.738608872949829905787237581602, −7.994657915612850532085512363979, −7.19469983756144866069776422875, −6.27524142638723587758777039836, −5.49082337578652377511534661995, −4.80633495631196671790732566421, −3.87272604590951610166563519472, −3.00000551981932457831061526159, −1.58402970774824736194299280222, −0.16873122761560147146285969276,
0.16873122761560147146285969276, 1.58402970774824736194299280222, 3.00000551981932457831061526159, 3.87272604590951610166563519472, 4.80633495631196671790732566421, 5.49082337578652377511534661995, 6.27524142638723587758777039836, 7.19469983756144866069776422875, 7.994657915612850532085512363979, 8.738608872949829905787237581602
