L(s) = 1 | + 3-s − 5-s + 9-s − 11-s + 6·13-s − 15-s + 7·17-s − 5·19-s + 23-s + 25-s + 27-s − 5·29-s − 8·31-s − 33-s − 2·37-s + 6·39-s − 12·41-s + 11·43-s − 45-s + 8·47-s + 7·51-s − 11·53-s + 55-s − 5·57-s − 5·59-s − 7·61-s − 6·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.258·15-s + 1.69·17-s − 1.14·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.960·39-s − 1.87·41-s + 1.67·43-s − 0.149·45-s + 1.16·47-s + 0.980·51-s − 1.51·53-s + 0.134·55-s − 0.662·57-s − 0.650·59-s − 0.896·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.507244859\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.507244859\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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
−13.49036291968107, −13.00537149855595, −12.52629793959071, −12.25712807975913, −11.47603110599423, −10.98347332953436, −10.66976353487557, −10.18686959161175, −9.506729744064677, −8.952182529732509, −8.674199732156838, −8.080220267662507, −7.632605235752031, −7.253070018384332, −6.550021632477032, −5.877090818357483, −5.633350550366766, −4.842626164981107, −4.163537592087087, −3.654336597711694, −3.334729559712060, −2.697499140866580, −1.717479414344698, −1.441741476742168, −0.4584973356715810,
0.4584973356715810, 1.441741476742168, 1.717479414344698, 2.697499140866580, 3.334729559712060, 3.654336597711694, 4.163537592087087, 4.842626164981107, 5.633350550366766, 5.877090818357483, 6.550021632477032, 7.253070018384332, 7.632605235752031, 8.080220267662507, 8.674199732156838, 8.952182529732509, 9.506729744064677, 10.18686959161175, 10.66976353487557, 10.98347332953436, 11.47603110599423, 12.25712807975913, 12.52629793959071, 13.00537149855595, 13.49036291968107