| L(s) = 1 | − 4·3-s + 2·5-s − 11·9-s + 44·11-s − 22·13-s − 8·15-s − 50·17-s + 44·19-s + 56·23-s − 121·25-s + 152·27-s + 198·29-s − 160·31-s − 176·33-s − 162·37-s + 88·39-s + 198·41-s − 52·43-s − 22·45-s + 528·47-s + 200·51-s − 242·53-s + 88·55-s − 176·57-s − 668·59-s − 550·61-s − 44·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.178·5-s − 0.407·9-s + 1.20·11-s − 0.469·13-s − 0.137·15-s − 0.713·17-s + 0.531·19-s + 0.507·23-s − 0.967·25-s + 1.08·27-s + 1.26·29-s − 0.926·31-s − 0.928·33-s − 0.719·37-s + 0.361·39-s + 0.754·41-s − 0.184·43-s − 0.0728·45-s + 1.63·47-s + 0.549·51-s − 0.627·53-s + 0.215·55-s − 0.408·57-s − 1.47·59-s − 1.15·61-s − 0.0839·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 162 T + p^{3} T^{2} \) |
| 41 | \( 1 - 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 242 T + p^{3} T^{2} \) |
| 59 | \( 1 + 668 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 728 T + p^{3} T^{2} \) |
| 73 | \( 1 + 154 T + p^{3} T^{2} \) |
| 79 | \( 1 - 656 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 + 714 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
| 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
−9.374678929129614221412052914191, −8.856665068709210199922643360434, −7.62622683440126051190839342766, −6.67312837193659420704866554694, −5.98866124186556696428845030890, −5.06321636358969952852239906619, −4.07902915854288173432026733261, −2.78161930979285828500314655002, −1.35696926398637994243540140334, 0,
1.35696926398637994243540140334, 2.78161930979285828500314655002, 4.07902915854288173432026733261, 5.06321636358969952852239906619, 5.98866124186556696428845030890, 6.67312837193659420704866554694, 7.62622683440126051190839342766, 8.856665068709210199922643360434, 9.374678929129614221412052914191
