| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 9·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 18·10-s + 62·11-s + 12·12-s − 13·13-s − 14·14-s + 27·15-s + 16·16-s − 16·17-s + 18·18-s + 79·19-s + 36·20-s − 21·21-s + 124·22-s − 155·23-s + 24·24-s − 44·25-s − 26·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.804·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.569·10-s + 1.69·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.464·15-s + 1/4·16-s − 0.228·17-s + 0.235·18-s + 0.953·19-s + 0.402·20-s − 0.218·21-s + 1.20·22-s − 1.40·23-s + 0.204·24-s − 0.351·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.719788467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.719788467\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 16 T + p^{3} T^{2} \) |
| 19 | \( 1 - 79 T + p^{3} T^{2} \) |
| 23 | \( 1 + 155 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 243 T + p^{3} T^{2} \) |
| 37 | \( 1 - 412 T + p^{3} T^{2} \) |
| 41 | \( 1 + 406 T + p^{3} T^{2} \) |
| 43 | \( 1 + 103 T + p^{3} T^{2} \) |
| 47 | \( 1 - 429 T + p^{3} T^{2} \) |
| 53 | \( 1 + 169 T + p^{3} T^{2} \) |
| 59 | \( 1 - 320 T + p^{3} T^{2} \) |
| 61 | \( 1 + 614 T + p^{3} T^{2} \) |
| 67 | \( 1 - 258 T + p^{3} T^{2} \) |
| 71 | \( 1 + 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 121 T + p^{3} T^{2} \) |
| 79 | \( 1 + 967 T + p^{3} T^{2} \) |
| 83 | \( 1 + 679 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1059 T + p^{3} T^{2} \) |
| 97 | \( 1 + 21 T + p^{3} T^{2} \) |
| 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
−10.09676875081580879783302892932, −9.687301052017913627540667414404, −8.727059813739549369961785687929, −7.56853505192019611412712318255, −6.49939869250736506745519092691, −5.94682664835536978361966910708, −4.56063371766411552206021481573, −3.65174307837280419107596728716, −2.49508048285391660506775410016, −1.33752500240726781740567136295,
1.33752500240726781740567136295, 2.49508048285391660506775410016, 3.65174307837280419107596728716, 4.56063371766411552206021481573, 5.94682664835536978361966910708, 6.49939869250736506745519092691, 7.56853505192019611412712318255, 8.727059813739549369961785687929, 9.687301052017913627540667414404, 10.09676875081580879783302892932
