L(s) = 1 | + 2·2-s + 2·4-s + 3·5-s + 6·10-s + 6·11-s + 13-s − 4·16-s − 17-s + 19-s + 6·20-s + 12·22-s + 5·23-s + 4·25-s + 2·26-s + 29-s − 31-s − 8·32-s − 2·34-s + 2·37-s + 2·38-s − 6·41-s − 5·43-s + 12·44-s + 10·46-s + 7·47-s + 8·50-s + 2·52-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 1.34·5-s + 1.89·10-s + 1.80·11-s + 0.277·13-s − 16-s − 0.242·17-s + 0.229·19-s + 1.34·20-s + 2.55·22-s + 1.04·23-s + 4/5·25-s + 0.392·26-s + 0.185·29-s − 0.179·31-s − 1.41·32-s − 0.342·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.762·43-s + 1.80·44-s + 1.47·46-s + 1.02·47-s + 1.13·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.489939587\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.489939587\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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
−13.81512884120136, −13.42083828070259, −12.86509096220604, −12.56818215734901, −11.80656713567626, −11.59195727089741, −11.04005944999795, −10.37696982317567, −9.785828464357407, −9.340490791512322, −8.831776509550640, −8.583713128742046, −7.401278030560699, −6.931401402661325, −6.473041050052429, −5.976998205184062, −5.723560924117947, −4.936188121913148, −4.554539530216830, −3.955884756496708, −3.256143328873008, −2.927500154071199, −1.942160536402570, −1.635308555159461, −0.7559223137832555,
0.7559223137832555, 1.635308555159461, 1.942160536402570, 2.927500154071199, 3.256143328873008, 3.955884756496708, 4.554539530216830, 4.936188121913148, 5.723560924117947, 5.976998205184062, 6.473041050052429, 6.931401402661325, 7.401278030560699, 8.583713128742046, 8.831776509550640, 9.340490791512322, 9.785828464357407, 10.37696982317567, 11.04005944999795, 11.59195727089741, 11.80656713567626, 12.56818215734901, 12.86509096220604, 13.42083828070259, 13.81512884120136