| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s − 5·13-s + 14-s − 16-s − 7·17-s − 6·19-s − 4·23-s + 5·26-s + 28-s − 9·29-s − 2·31-s − 5·32-s + 7·34-s − 9·37-s + 6·38-s + 7·41-s + 6·43-s + 4·46-s + 2·47-s + 49-s + 5·52-s + 3·53-s − 3·56-s + 9·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 1.38·13-s + 0.267·14-s − 1/4·16-s − 1.69·17-s − 1.37·19-s − 0.834·23-s + 0.980·26-s + 0.188·28-s − 1.67·29-s − 0.359·31-s − 0.883·32-s + 1.20·34-s − 1.47·37-s + 0.973·38-s + 1.09·41-s + 0.914·43-s + 0.589·46-s + 0.291·47-s + 1/7·49-s + 0.693·52-s + 0.412·53-s − 0.400·56-s + 1.18·58-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.27877319557244, −12.90823182242131, −12.45165771879485, −12.03937869213264, −11.26218472108212, −10.84291051913568, −10.51157372383370, −9.953893170655980, −9.465313413403021, −9.114625181136441, −8.684797267664824, −8.261457758855412, −7.480869966280333, −7.302400853149311, −6.751212998398048, −6.087401374701722, −5.559641690839452, −4.962064839673067, −4.276967111675071, −4.173190813838153, −3.446109634726911, −2.386873730463670, −2.215865565602481, −1.544341901515605, −0.4027272667655672, 0,
0.4027272667655672, 1.544341901515605, 2.215865565602481, 2.386873730463670, 3.446109634726911, 4.173190813838153, 4.276967111675071, 4.962064839673067, 5.559641690839452, 6.087401374701722, 6.751212998398048, 7.302400853149311, 7.480869966280333, 8.261457758855412, 8.684797267664824, 9.114625181136441, 9.465313413403021, 9.953893170655980, 10.51157372383370, 10.84291051913568, 11.26218472108212, 12.03937869213264, 12.45165771879485, 12.90823182242131, 13.27877319557244
