| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − 9-s + 2·11-s − i·12-s + i·13-s + 14-s + 16-s + i·17-s + i·18-s − 4·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + 0.603·11-s − 0.288i·12-s + 0.277i·13-s + 0.267·14-s + 0.250·16-s + 0.242i·17-s + 0.235i·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.210251011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210251011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 7iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 11iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
| 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
−10.04302641958911688138489255893, −9.338761306991237855066074074364, −8.715287919076421159093313192164, −7.79344127461355736747901157906, −6.53313358721282531834487284066, −5.64584033056976324451553667184, −4.60667921129231490340718220403, −3.82328573754643531973180688230, −2.79356607504151524702706189280, −1.51460859731706038790523289467,
0.56606977179758571944245811811, 2.17852028862336291800327597486, 3.63359521919827120895862139312, 4.61219213840224294215808510770, 5.65529872225352728739475029134, 6.65087616263192343300580267071, 7.00352492793727908429565553928, 8.190949308105722393820841666077, 8.594499813768740827429644427057, 9.664948354568831533998112642834
