L(s) = 1 | − 3-s − 5-s − 3.13·7-s + 9-s − 2.65·11-s + 0.651·13-s + 15-s + 4.48·17-s + 0.651·19-s + 3.13·21-s − 23-s + 25-s − 27-s + 2.48·29-s + 3.13·31-s + 2.65·33-s + 3.13·35-s − 6.10·37-s − 0.651·39-s + 0.820·41-s − 0.696·43-s − 45-s + 11.8·47-s + 2.83·49-s − 4.48·51-s + 9.45·53-s + 2.65·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.18·7-s + 0.333·9-s − 0.799·11-s + 0.180·13-s + 0.258·15-s + 1.08·17-s + 0.149·19-s + 0.684·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.461·29-s + 0.563·31-s + 0.461·33-s + 0.530·35-s − 1.00·37-s − 0.104·39-s + 0.128·41-s − 0.106·43-s − 0.149·45-s + 1.73·47-s + 0.404·49-s − 0.627·51-s + 1.29·53-s + 0.357·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 - 0.651T + 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 - 0.651T + 19T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 - 0.820T + 41T^{2} \) |
| 43 | \( 1 + 0.696T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 9.45T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 6.31T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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
−7.66845488519256044728747410436, −7.09436835935119714163296833583, −6.32069073048154739223890032847, −5.68131607578550732693994556249, −5.00604404174352767014504480294, −4.01870177500749694701357688897, −3.31266233979619403431868187756, −2.52075107973109547310492990921, −1.05770302617240638448708093010, 0,
1.05770302617240638448708093010, 2.52075107973109547310492990921, 3.31266233979619403431868187756, 4.01870177500749694701357688897, 5.00604404174352767014504480294, 5.68131607578550732693994556249, 6.32069073048154739223890032847, 7.09436835935119714163296833583, 7.66845488519256044728747410436