| L(s) = 1 | + 5-s − 7-s − 2·13-s + 2·19-s + 25-s − 6·29-s + 4·31-s − 35-s − 4·37-s + 9·41-s − 43-s − 3·47-s − 6·49-s + 6·53-s + 61-s − 2·65-s + 13·67-s − 12·71-s + 16·73-s − 10·79-s − 12·83-s + 3·89-s + 2·91-s + 2·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.554·13-s + 0.458·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.657·37-s + 1.40·41-s − 0.152·43-s − 0.437·47-s − 6/7·49-s + 0.824·53-s + 0.128·61-s − 0.248·65-s + 1.58·67-s − 1.42·71-s + 1.87·73-s − 1.12·79-s − 1.31·83-s + 0.317·89-s + 0.209·91-s + 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.10899950912650, −13.66960336756105, −13.11599973781356, −12.74703215646351, −12.22762156325579, −11.70413689589792, −11.15525389969186, −10.69831501831522, −10.04703407640394, −9.605709429576414, −9.371013789055391, −8.621668207632298, −8.116160308863210, −7.542097497600143, −6.921126994723563, −6.626341471688847, −5.716852531154896, −5.580808092854451, −4.810463014100404, −4.240416783264914, −3.562810658072527, −2.947504761343161, −2.361588688439998, −1.695234784158665, −0.8912516605150102, 0,
0.8912516605150102, 1.695234784158665, 2.361588688439998, 2.947504761343161, 3.562810658072527, 4.240416783264914, 4.810463014100404, 5.580808092854451, 5.716852531154896, 6.626341471688847, 6.921126994723563, 7.542097497600143, 8.116160308863210, 8.621668207632298, 9.371013789055391, 9.605709429576414, 10.04703407640394, 10.69831501831522, 11.15525389969186, 11.70413689589792, 12.22762156325579, 12.74703215646351, 13.11599973781356, 13.66960336756105, 14.10899950912650
