| L(s) = 1 | − 2·2-s + 2·4-s − 3·5-s + 6·10-s − 2·11-s − 13-s − 4·16-s + 17-s − 19-s − 6·20-s + 4·22-s − 3·23-s + 4·25-s + 2·26-s − 7·29-s + 5·31-s + 8·32-s − 2·34-s + 6·37-s + 2·38-s + 6·41-s + 11·43-s − 4·44-s + 6·46-s + 5·47-s − 8·50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.34·5-s + 1.89·10-s − 0.603·11-s − 0.277·13-s − 16-s + 0.242·17-s − 0.229·19-s − 1.34·20-s + 0.852·22-s − 0.625·23-s + 4/5·25-s + 0.392·26-s − 1.29·29-s + 0.898·31-s + 1.41·32-s − 0.342·34-s + 0.986·37-s + 0.324·38-s + 0.937·41-s + 1.67·43-s − 0.603·44-s + 0.884·46-s + 0.729·47-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
| \(L(1)\) |
\(\approx\) |
\(0.5672695153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5672695153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.87404577220165, −13.11834742812820, −12.73794348947713, −12.12217926716007, −11.67898875314258, −11.19337827717718, −10.72859194268646, −10.42242529373046, −9.563880042381450, −9.469708282409816, −8.689798390687644, −8.285885291660858, −7.687954097968409, −7.568422948618657, −7.135439344096510, −6.238913245175758, −5.798184771421289, −4.925009290401063, −4.289172282532647, −4.014080062355272, −3.109207863547457, −2.485732821157619, −1.848419480442819, −0.8623981506274456, −0.4053576966910275,
0.4053576966910275, 0.8623981506274456, 1.848419480442819, 2.485732821157619, 3.109207863547457, 4.014080062355272, 4.289172282532647, 4.925009290401063, 5.798184771421289, 6.238913245175758, 7.135439344096510, 7.568422948618657, 7.687954097968409, 8.285885291660858, 8.689798390687644, 9.469708282409816, 9.563880042381450, 10.42242529373046, 10.72859194268646, 11.19337827717718, 11.67898875314258, 12.12217926716007, 12.73794348947713, 13.11834742812820, 13.87404577220165
