L(s) = 1 | − 3·7-s − 11-s + 2·13-s − 3·17-s − 19-s + 23-s + 6·29-s − 4·31-s + 37-s + 5·41-s − 4·43-s + 3·47-s + 2·49-s + 10·53-s − 11·59-s − 14·61-s − 2·67-s − 5·71-s − 2·73-s + 3·77-s − 5·79-s − 8·83-s + 10·89-s − 6·91-s + 17·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 0.301·11-s + 0.554·13-s − 0.727·17-s − 0.229·19-s + 0.208·23-s + 1.11·29-s − 0.718·31-s + 0.164·37-s + 0.780·41-s − 0.609·43-s + 0.437·47-s + 2/7·49-s + 1.37·53-s − 1.43·59-s − 1.79·61-s − 0.244·67-s − 0.593·71-s − 0.234·73-s + 0.341·77-s − 0.562·79-s − 0.878·83-s + 1.05·89-s − 0.628·91-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9851593581\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9851593581\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.19518602286537, −12.92582882964225, −12.39850955902353, −11.91901767440827, −11.40974475833787, −10.74855608915920, −10.48163988319205, −10.04286385897467, −9.287039145626583, −9.059515682649178, −8.617083323857556, −7.874758073116181, −7.509796157745060, −6.712382305569061, −6.564043602605036, −5.916343145582617, −5.522268146629862, −4.698015871383049, −4.290250683694817, −3.672198101181845, −3.014980831294849, −2.700310979279900, −1.890204190671355, −1.157626073486213, −0.3042589993629453,
0.3042589993629453, 1.157626073486213, 1.890204190671355, 2.700310979279900, 3.014980831294849, 3.672198101181845, 4.290250683694817, 4.698015871383049, 5.522268146629862, 5.916343145582617, 6.564043602605036, 6.712382305569061, 7.509796157745060, 7.874758073116181, 8.617083323857556, 9.059515682649178, 9.287039145626583, 10.04286385897467, 10.48163988319205, 10.74855608915920, 11.40974475833787, 11.91901767440827, 12.39850955902353, 12.92582882964225, 13.19518602286537