L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 6·13-s + 14-s − 16-s − 2·17-s − 4·19-s + 6·26-s − 28-s − 2·29-s + 8·31-s + 5·32-s − 2·34-s − 6·37-s − 4·38-s + 10·41-s − 4·43-s − 8·47-s + 49-s − 6·52-s + 6·53-s − 3·56-s − 2·58-s − 4·59-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 1.17·26-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.832·52-s + 0.824·53-s − 0.400·56-s − 0.262·58-s − 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.901192091\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.901192091\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−13.08864137230128, −12.75629747642137, −12.40675169911174, −11.52832816996748, −11.40817740195166, −10.95514704097447, −10.16931325189279, −9.972666010021363, −9.167346940518820, −8.738325902026065, −8.372467750564006, −8.120999930558303, −7.226032738012645, −6.616594175022908, −6.267719634523544, −5.726496464761837, −5.302140345348989, −4.584464594189375, −4.255748000749817, −3.773239998898003, −3.234579439771950, −2.588697066070746, −1.889660361296940, −1.157869521450271, −0.4677671949628428,
0.4677671949628428, 1.157869521450271, 1.889660361296940, 2.588697066070746, 3.234579439771950, 3.773239998898003, 4.255748000749817, 4.584464594189375, 5.302140345348989, 5.726496464761837, 6.267719634523544, 6.616594175022908, 7.226032738012645, 8.120999930558303, 8.372467750564006, 8.738325902026065, 9.167346940518820, 9.972666010021363, 10.16931325189279, 10.95514704097447, 11.40817740195166, 11.52832816996748, 12.40675169911174, 12.75629747642137, 13.08864137230128