L(s) = 1 | + 3-s + 9-s − 5·11-s + 2·13-s + 6·17-s + 2·19-s + 5·23-s + 27-s + 5·29-s − 4·31-s − 5·33-s + 37-s + 2·39-s + 12·41-s − 5·43-s − 2·47-s + 6·51-s + 14·53-s + 2·57-s − 2·59-s + 5·67-s + 5·69-s + 9·71-s + 10·73-s − 11·79-s + 81-s + 16·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s + 1.04·23-s + 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.870·33-s + 0.164·37-s + 0.320·39-s + 1.87·41-s − 0.762·43-s − 0.291·47-s + 0.840·51-s + 1.92·53-s + 0.264·57-s − 0.260·59-s + 0.610·67-s + 0.601·69-s + 1.06·71-s + 1.17·73-s − 1.23·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.725580226\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.725580226\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−13.03974210135392, −12.56774074703719, −12.06446718887321, −11.49050360959853, −10.99001306841345, −10.49429171942341, −10.15300239503607, −9.703127830689690, −9.084846677433723, −8.724979741971687, −8.111734581731431, −7.711370253915804, −7.453102644247226, −6.826035927658777, −6.163921564361606, −5.640669237739578, −5.141538734021220, −4.808434718884130, −4.008189182908257, −3.355180381660329, −3.141966125069906, −2.421206903104403, −1.979544384602657, −0.9264519113286336, −0.7329502818576649,
0.7329502818576649, 0.9264519113286336, 1.979544384602657, 2.421206903104403, 3.141966125069906, 3.355180381660329, 4.008189182908257, 4.808434718884130, 5.141538734021220, 5.640669237739578, 6.163921564361606, 6.826035927658777, 7.453102644247226, 7.711370253915804, 8.111734581731431, 8.724979741971687, 9.084846677433723, 9.703127830689690, 10.15300239503607, 10.49429171942341, 10.99001306841345, 11.49050360959853, 12.06446718887321, 12.56774074703719, 13.03974210135392