| L(s) = 1 | − 2·4-s − 6·11-s + 4·16-s − 6·17-s − 19-s + 6·23-s − 5·25-s − 6·29-s + 5·31-s + 37-s − 4·43-s + 12·44-s − 6·47-s − 6·53-s + 13·61-s − 8·64-s + 13·67-s + 12·68-s − 12·71-s − 10·73-s + 2·76-s − 79-s + 6·83-s − 6·89-s − 12·92-s + 5·97-s + 10·100-s + ⋯ |
| L(s) = 1 | − 4-s − 1.80·11-s + 16-s − 1.45·17-s − 0.229·19-s + 1.25·23-s − 25-s − 1.11·29-s + 0.898·31-s + 0.164·37-s − 0.609·43-s + 1.80·44-s − 0.875·47-s − 0.824·53-s + 1.66·61-s − 64-s + 1.58·67-s + 1.45·68-s − 1.42·71-s − 1.17·73-s + 0.229·76-s − 0.112·79-s + 0.658·83-s − 0.635·89-s − 1.25·92-s + 0.507·97-s + 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.23738762983312, −13.69529518018567, −13.14799902665268, −12.95555320305804, −12.76204136657652, −11.63051916851135, −11.40280330629871, −10.74150011630435, −10.22167835131183, −9.845478689398321, −9.234128679772794, −8.678560843681939, −8.309711753721687, −7.735391969814359, −7.273189508183045, −6.535991121051209, −5.905347986993419, −5.297633358045787, −4.863131488752090, −4.441617511712274, −3.691815589758945, −3.063438779068360, −2.401024086543341, −1.744684213756959, −0.6216911680090648, 0,
0.6216911680090648, 1.744684213756959, 2.401024086543341, 3.063438779068360, 3.691815589758945, 4.441617511712274, 4.863131488752090, 5.297633358045787, 5.905347986993419, 6.535991121051209, 7.273189508183045, 7.735391969814359, 8.309711753721687, 8.678560843681939, 9.234128679772794, 9.845478689398321, 10.22167835131183, 10.74150011630435, 11.40280330629871, 11.63051916851135, 12.76204136657652, 12.95555320305804, 13.14799902665268, 13.69529518018567, 14.23738762983312
