L(s) = 1 | − 3-s + 9-s + 6·11-s − 13-s − 3·17-s + 4·19-s + 3·23-s − 27-s − 3·29-s + 5·31-s − 6·33-s + 10·37-s + 39-s − 9·41-s − 43-s + 3·51-s − 9·53-s − 4·57-s − 9·59-s + 11·61-s − 4·67-s − 3·69-s + 12·71-s + 10·73-s + 10·79-s + 81-s − 9·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.80·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s + 0.625·23-s − 0.192·27-s − 0.557·29-s + 0.898·31-s − 1.04·33-s + 1.64·37-s + 0.160·39-s − 1.40·41-s − 0.152·43-s + 0.420·51-s − 1.23·53-s − 0.529·57-s − 1.17·59-s + 1.40·61-s − 0.488·67-s − 0.361·69-s + 1.42·71-s + 1.17·73-s + 1.12·79-s + 1/9·81-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.643456125\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.643456125\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.81839576266081, −12.37088992545227, −11.90003135029057, −11.50745203800990, −11.18849977005457, −10.77144944881547, −9.985363351148373, −9.572815046300739, −9.359378764571290, −8.780028040590361, −8.163696944126345, −7.729814842006905, −6.899771531221196, −6.827547277719122, −6.269702617977023, −5.803874658150116, −5.113490495678135, −4.671731117737893, −4.222067491894227, −3.581892750864101, −3.145259681767742, −2.324443705014300, −1.671168370753898, −1.087915648828132, −0.5290392800752154,
0.5290392800752154, 1.087915648828132, 1.671168370753898, 2.324443705014300, 3.145259681767742, 3.581892750864101, 4.222067491894227, 4.671731117737893, 5.113490495678135, 5.803874658150116, 6.269702617977023, 6.827547277719122, 6.899771531221196, 7.729814842006905, 8.163696944126345, 8.780028040590361, 9.359378764571290, 9.572815046300739, 9.985363351148373, 10.77144944881547, 11.18849977005457, 11.50745203800990, 11.90003135029057, 12.37088992545227, 12.81839576266081