| L(s) = 1 | − 0.604·3-s − 3.83·5-s + 7-s − 2.63·9-s + 5.52·11-s + 0.106·13-s + 2.31·15-s − 0.497·17-s + 1.82·19-s − 0.604·21-s + 23-s + 9.73·25-s + 3.40·27-s + 7.25·29-s + 3.94·31-s − 3.34·33-s − 3.83·35-s + 5.37·37-s − 0.0642·39-s + 5.10·41-s + 2.40·43-s + 10.1·45-s − 8.35·47-s + 49-s + 0.300·51-s − 8.25·53-s − 21.2·55-s + ⋯ |
| L(s) = 1 | − 0.348·3-s − 1.71·5-s + 0.377·7-s − 0.878·9-s + 1.66·11-s + 0.0294·13-s + 0.598·15-s − 0.120·17-s + 0.419·19-s − 0.131·21-s + 0.208·23-s + 1.94·25-s + 0.655·27-s + 1.34·29-s + 0.708·31-s − 0.581·33-s − 0.648·35-s + 0.884·37-s − 0.0102·39-s + 0.796·41-s + 0.367·43-s + 1.50·45-s − 1.21·47-s + 0.142·49-s + 0.0421·51-s − 1.13·53-s − 2.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9892188611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9892188611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.604T + 3T^{2} \) |
| 5 | \( 1 + 3.83T + 5T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 - 0.106T + 13T^{2} \) |
| 17 | \( 1 + 0.497T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 - 3.94T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 - 5.10T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 + 8.35T + 47T^{2} \) |
| 53 | \( 1 + 8.25T + 53T^{2} \) |
| 59 | \( 1 - 0.668T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 0.897T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−11.01275872144416945096840726345, −9.621121405795724559698332200831, −8.587933493752205253635954921494, −8.086692866449558084304282644422, −7.01080017760588942468651425387, −6.21345520013373392514239248476, −4.82713631504885421256949753974, −4.05381173462149372591988548171, −3.03189296623008967128935039623, −0.894531937057956758755581108434,
0.894531937057956758755581108434, 3.03189296623008967128935039623, 4.05381173462149372591988548171, 4.82713631504885421256949753974, 6.21345520013373392514239248476, 7.01080017760588942468651425387, 8.086692866449558084304282644422, 8.587933493752205253635954921494, 9.621121405795724559698332200831, 11.01275872144416945096840726345
