| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 13-s + 16-s − 8·17-s + 6·19-s − 4·22-s + 6·23-s − 26-s + 4·29-s + 32-s − 8·34-s + 2·37-s + 6·38-s − 2·41-s + 4·43-s − 4·44-s + 6·46-s − 52-s − 10·53-s + 4·58-s + 4·59-s + 10·61-s + 64-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 1.94·17-s + 1.37·19-s − 0.852·22-s + 1.25·23-s − 0.196·26-s + 0.742·29-s + 0.176·32-s − 1.37·34-s + 0.328·37-s + 0.973·38-s − 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.884·46-s − 0.138·52-s − 1.37·53-s + 0.525·58-s + 0.520·59-s + 1.28·61-s + 1/8·64-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.883413743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.883413743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.82957795721431, −12.43325129227241, −11.77299650569504, −11.36014912713838, −10.95349466812623, −10.66195852849267, −9.975335022935229, −9.578770568090461, −9.066829647966739, −8.426263387573786, −8.124568742393056, −7.421552057749081, −6.991848803696353, −6.767524309256667, −5.977669123148812, −5.538349457870025, −5.054982187942542, −4.596105882198885, −4.262916940867240, −3.417635711277960, −2.851664124029606, −2.626583655049382, −1.928086992311933, −1.190683876671855, −0.4020751437497275,
0.4020751437497275, 1.190683876671855, 1.928086992311933, 2.626583655049382, 2.851664124029606, 3.417635711277960, 4.262916940867240, 4.596105882198885, 5.054982187942542, 5.538349457870025, 5.977669123148812, 6.767524309256667, 6.991848803696353, 7.421552057749081, 8.124568742393056, 8.426263387573786, 9.066829647966739, 9.578770568090461, 9.975335022935229, 10.66195852849267, 10.95349466812623, 11.36014912713838, 11.77299650569504, 12.43325129227241, 12.82957795721431
