| L(s) = 1 | − 3-s + 4·7-s + 9-s + 2·11-s + 13-s − 2·19-s − 4·21-s − 2·23-s − 27-s + 10·29-s − 4·31-s − 2·33-s − 6·37-s − 39-s − 6·41-s + 8·43-s + 12·47-s + 9·49-s − 14·53-s + 2·57-s − 6·59-s + 2·61-s + 4·63-s − 4·67-s + 2·69-s + 14·73-s + 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.458·19-s − 0.872·21-s − 0.417·23-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s − 1.92·53-s + 0.264·57-s − 0.781·59-s + 0.256·61-s + 0.503·63-s − 0.488·67-s + 0.240·69-s + 1.63·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.487242099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.487242099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.14404279270847, −14.41658719342965, −14.05126990949393, −13.78002399177878, −12.76913996895832, −12.35292715025491, −11.85595555803188, −11.39550769047927, −10.78577725368913, −10.51588370151393, −9.817787116188416, −8.863863557569679, −8.759897947741988, −7.850968586425345, −7.582745874571350, −6.644045667555324, −6.319878471031639, −5.485195468831794, −5.011714888370237, −4.381296969870324, −3.930049411498191, −2.958650429965261, −1.991619495603084, −1.476788799446027, −0.6564176196834244,
0.6564176196834244, 1.476788799446027, 1.991619495603084, 2.958650429965261, 3.930049411498191, 4.381296969870324, 5.011714888370237, 5.485195468831794, 6.319878471031639, 6.644045667555324, 7.582745874571350, 7.850968586425345, 8.759897947741988, 8.863863557569679, 9.817787116188416, 10.51588370151393, 10.78577725368913, 11.39550769047927, 11.85595555803188, 12.35292715025491, 12.76913996895832, 13.78002399177878, 14.05126990949393, 14.41658719342965, 15.14404279270847
