| L(s) = 1 | + 2-s + 4-s + 8-s − 6·13-s + 16-s − 17-s − 8·23-s − 6·26-s + 6·29-s + 8·31-s + 32-s − 34-s − 10·37-s − 6·41-s − 12·43-s − 8·46-s − 6·52-s − 10·53-s + 6·58-s − 8·59-s − 6·61-s + 8·62-s + 64-s − 12·67-s − 68-s − 6·73-s − 10·74-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.66·13-s + 1/4·16-s − 0.242·17-s − 1.66·23-s − 1.17·26-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s − 1.64·37-s − 0.937·41-s − 1.82·43-s − 1.17·46-s − 0.832·52-s − 1.37·53-s + 0.787·58-s − 1.04·59-s − 0.768·61-s + 1.01·62-s + 1/8·64-s − 1.46·67-s − 0.121·68-s − 0.702·73-s − 1.16·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.77892805887034, −12.50919300775985, −11.93659185938807, −11.81002301136652, −11.43774501576435, −10.43924153446646, −10.25222613028124, −10.06169907602584, −9.411059187063362, −8.771734341036139, −8.324412484418282, −7.823233912847602, −7.446959835521062, −6.777992243058200, −6.525109296056781, −6.009373832959036, −5.383647986871662, −4.877202521250884, −4.556253078787958, −4.159400368112053, −3.196848163406994, −3.097183219345163, −2.377740020992834, −1.800346759819736, −1.350609905798230, 0, 0,
1.350609905798230, 1.800346759819736, 2.377740020992834, 3.097183219345163, 3.196848163406994, 4.159400368112053, 4.556253078787958, 4.877202521250884, 5.383647986871662, 6.009373832959036, 6.525109296056781, 6.777992243058200, 7.446959835521062, 7.823233912847602, 8.324412484418282, 8.771734341036139, 9.411059187063362, 10.06169907602584, 10.25222613028124, 10.43924153446646, 11.43774501576435, 11.81002301136652, 11.93659185938807, 12.50919300775985, 12.77892805887034
