L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 4·11-s − 6·13-s − 14-s + 16-s + 6·19-s + 4·22-s − 23-s − 6·26-s − 28-s − 8·29-s + 8·31-s + 32-s − 2·37-s + 6·38-s + 2·41-s − 8·43-s + 4·44-s − 46-s + 12·47-s + 49-s − 6·52-s − 2·53-s − 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.852·22-s − 0.208·23-s − 1.17·26-s − 0.188·28-s − 1.48·29-s + 1.43·31-s + 0.176·32-s − 0.328·37-s + 0.973·38-s + 0.312·41-s − 1.21·43-s + 0.603·44-s − 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.832·52-s − 0.274·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.09838980504376, −14.02750652070822, −13.46704010136012, −12.80090543396725, −12.25474853937219, −11.91774671478528, −11.67682377392597, −10.91453373645173, −10.32457086218236, −9.749985312455648, −9.367757296753802, −8.981636656630915, −7.946724060940014, −7.644598697909222, −6.970696873854962, −6.683108646722118, −5.923453277061285, −5.457324774535251, −4.843750859161138, −4.313208837085370, −3.702422298325287, −3.092592771195979, −2.547223913501232, −1.772645183702904, −1.046812029357416, 0,
1.046812029357416, 1.772645183702904, 2.547223913501232, 3.092592771195979, 3.702422298325287, 4.313208837085370, 4.843750859161138, 5.457324774535251, 5.923453277061285, 6.683108646722118, 6.970696873854962, 7.644598697909222, 7.946724060940014, 8.981636656630915, 9.367757296753802, 9.749985312455648, 10.32457086218236, 10.91453373645173, 11.67682377392597, 11.91774671478528, 12.25474853937219, 12.80090543396725, 13.46704010136012, 14.02750652070822, 14.09838980504376