| L(s) = 1 | + 3-s + 2·7-s + 9-s + 6·11-s + 13-s + 17-s + 2·19-s + 2·21-s − 6·23-s + 27-s − 9·29-s − 9·31-s + 6·33-s + 12·37-s + 39-s + 6·41-s + 6·43-s − 12·47-s − 3·49-s + 51-s + 3·53-s + 2·57-s + 7·59-s − 6·61-s + 2·63-s − 13·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.242·17-s + 0.458·19-s + 0.436·21-s − 1.25·23-s + 0.192·27-s − 1.67·29-s − 1.61·31-s + 1.04·33-s + 1.97·37-s + 0.160·39-s + 0.937·41-s + 0.914·43-s − 1.75·47-s − 3/7·49-s + 0.140·51-s + 0.412·53-s + 0.264·57-s + 0.911·59-s − 0.768·61-s + 0.251·63-s − 1.58·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.459741732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.459741732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.53410825074150, −12.95801129826902, −12.63522993631787, −11.82231036204110, −11.61766165324167, −11.15650837628558, −10.68829416978127, −9.857774891921392, −9.468888170254132, −9.180078612323340, −8.649752998942411, −8.036223948241824, −7.521469464359518, −7.316415285337346, −6.449953190835735, −5.967454820485921, −5.617055180026601, −4.668035005448284, −4.298625701141221, −3.674609677373130, −3.421434519757889, −2.436516504350145, −1.759206897470840, −1.471489734399226, −0.6141697570102893,
0.6141697570102893, 1.471489734399226, 1.759206897470840, 2.436516504350145, 3.421434519757889, 3.674609677373130, 4.298625701141221, 4.668035005448284, 5.617055180026601, 5.967454820485921, 6.449953190835735, 7.316415285337346, 7.521469464359518, 8.036223948241824, 8.649752998942411, 9.180078612323340, 9.468888170254132, 9.857774891921392, 10.68829416978127, 11.15650837628558, 11.61766165324167, 11.82231036204110, 12.63522993631787, 12.95801129826902, 13.53410825074150
