| L(s) = 1 | − 3i·3-s + 7i·7-s − 9·9-s − 36·11-s + 34i·13-s − 30i·17-s + 16·19-s + 21·21-s + 48i·23-s + 27i·27-s + 126·29-s + 8·31-s + 108i·33-s + 74i·37-s + 102·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 0.986·11-s + 0.725i·13-s − 0.428i·17-s + 0.193·19-s + 0.218·21-s + 0.435i·23-s + 0.192i·27-s + 0.806·29-s + 0.0463·31-s + 0.569i·33-s + 0.328i·37-s + 0.418·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.025558792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025558792\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 11 | \( 1 + 36T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 30iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 16T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 126T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 74iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 138T + 6.89e4T^{2} \) |
| 43 | \( 1 - 352iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 396iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 78iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 60T + 2.05e5T^{2} \) |
| 61 | \( 1 + 70T + 2.26e5T^{2} \) |
| 67 | \( 1 + 664iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 156T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 344T + 4.93e5T^{2} \) |
| 83 | \( 1 + 444iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 290iT - 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.393267441777175905907962778155, −7.74916169088498789544718434150, −6.95178606517792490910209131183, −6.21467586969407099951190361346, −5.31546744919023550959294814484, −4.60801692950454582996349347508, −3.31137782658847895757978248504, −2.48667486002917377953978560383, −1.52885596641496405473955220661, −0.24576999120355915561151378700,
0.883685517633251177104400200760, 2.35619306547183797647090020864, 3.21677634159326772142409375770, 4.13807422858179388998563111001, 5.02210729344885638894276885703, 5.67792282955767409232022091934, 6.63510164473088108844196798724, 7.59647083240353638400478989481, 8.229213327560137911543751965094, 8.957936908651738006872242812487
