| L(s) = 1 | − 2-s + 4-s − 8-s − 4·11-s − 13-s + 16-s − 2·17-s − 4·19-s + 4·22-s + 4·23-s + 26-s − 2·29-s + 8·31-s − 32-s + 2·34-s + 6·37-s + 4·38-s + 10·41-s + 4·43-s − 4·44-s − 4·46-s − 52-s − 6·53-s + 2·58-s + 8·59-s − 10·61-s − 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.852·22-s + 0.834·23-s + 0.196·26-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.986·37-s + 0.648·38-s + 1.56·41-s + 0.609·43-s − 0.603·44-s − 0.589·46-s − 0.138·52-s − 0.824·53-s + 0.262·58-s + 1.04·59-s − 1.28·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.84856334499602, −12.59667129637141, −12.05169978750440, −11.31470263395758, −11.00885412333510, −10.80215761022646, −10.11661292542106, −9.748665899839555, −9.328794111565324, −8.754149042959112, −8.306697801086474, −7.892530103800852, −7.485310655608636, −6.967735283167278, −6.304690854835574, −6.133033081965729, −5.269451120575194, −4.982828147309023, −4.280944408242020, −3.849841055967139, −2.896869153968081, −2.571730059260513, −2.226826254126432, −1.307064616202099, −0.6922067687119505, 0,
0.6922067687119505, 1.307064616202099, 2.226826254126432, 2.571730059260513, 2.896869153968081, 3.849841055967139, 4.280944408242020, 4.982828147309023, 5.269451120575194, 6.133033081965729, 6.304690854835574, 6.967735283167278, 7.485310655608636, 7.892530103800852, 8.306697801086474, 8.754149042959112, 9.328794111565324, 9.748665899839555, 10.11661292542106, 10.80215761022646, 11.00885412333510, 11.31470263395758, 12.05169978750440, 12.59667129637141, 12.84856334499602
