| L(s) = 1 | − 7-s − 3·9-s + 4·13-s + 4·17-s + 2·19-s − 23-s − 6·29-s − 2·31-s + 6·37-s − 2·41-s + 4·43-s + 6·47-s + 49-s − 6·53-s − 2·61-s + 3·63-s − 4·67-s − 12·71-s + 10·73-s + 8·79-s + 9·81-s − 6·83-s − 4·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.10·13-s + 0.970·17-s + 0.458·19-s − 0.208·23-s − 1.11·29-s − 0.359·31-s + 0.986·37-s − 0.312·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 0.256·61-s + 0.377·63-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 81-s − 0.658·83-s − 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.45819705624728, −13.90547913950203, −13.61700355006525, −12.92906774104465, −12.56356821442289, −11.79927690665265, −11.58867675663214, −10.83653307424485, −10.63578267170982, −9.818749133966215, −9.267148062403761, −9.008246433569132, −8.213061669606345, −7.861199051958494, −7.301348573917947, −6.539116773627206, −5.985633391789955, −5.641079940019052, −5.119939487376733, −4.161717256627177, −3.684185515579016, −3.101481490766317, −2.558801931792168, −1.628899506949462, −0.9231239484488332, 0,
0.9231239484488332, 1.628899506949462, 2.558801931792168, 3.101481490766317, 3.684185515579016, 4.161717256627177, 5.119939487376733, 5.641079940019052, 5.985633391789955, 6.539116773627206, 7.301348573917947, 7.861199051958494, 8.213061669606345, 9.008246433569132, 9.267148062403761, 9.818749133966215, 10.63578267170982, 10.83653307424485, 11.58867675663214, 11.79927690665265, 12.56356821442289, 12.92906774104465, 13.61700355006525, 13.90547913950203, 14.45819705624728
