L(s) = 1 | − 7-s − 6·13-s + 2·17-s + 8·19-s + 8·23-s − 2·29-s + 4·31-s − 2·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 10·53-s + 4·59-s + 2·61-s + 4·67-s + 12·71-s + 2·73-s + 8·79-s + 4·83-s + 6·89-s + 6·91-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 1.66·23-s − 0.371·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s + 0.439·83-s + 0.635·89-s + 0.628·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.777274232\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.777274232\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.80862564973686, −13.22474963402418, −12.67891431367522, −12.31671699049049, −11.88706543266123, −11.32333857594726, −10.83004289406527, −10.22373826560904, −9.664877255146692, −9.413142164677854, −8.994332723105635, −8.134383148265220, −7.599528595672463, −7.273215009082166, −6.837155589899355, −6.093114220547402, −5.475309487965591, −5.000371468317004, −4.665539116442852, −3.713265901050085, −3.217017703284143, −2.672808653572593, −2.114792124481348, −1.033494391099619, −0.6241286256797853,
0.6241286256797853, 1.033494391099619, 2.114792124481348, 2.672808653572593, 3.217017703284143, 3.713265901050085, 4.665539116442852, 5.000371468317004, 5.475309487965591, 6.093114220547402, 6.837155589899355, 7.273215009082166, 7.599528595672463, 8.134383148265220, 8.994332723105635, 9.413142164677854, 9.664877255146692, 10.22373826560904, 10.83004289406527, 11.32333857594726, 11.88706543266123, 12.31671699049049, 12.67891431367522, 13.22474963402418, 13.80862564973686