| L(s) = 1 | − 2i·3-s + (−1 + 2i)5-s + 2i·7-s − 9-s − 4·11-s − 4i·13-s + (4 + 2i)15-s + 4·19-s + 4·21-s + 2i·23-s + (−3 − 4i)25-s − 4i·27-s − 2·29-s + 8i·33-s + (−4 − 2i)35-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (−0.447 + 0.894i)5-s + 0.755i·7-s − 0.333·9-s − 1.20·11-s − 1.10i·13-s + (1.03 + 0.516i)15-s + 0.917·19-s + 0.872·21-s + 0.417i·23-s + (−0.600 − 0.800i)25-s − 0.769i·27-s − 0.371·29-s + 1.39i·33-s + (−0.676 − 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.712587 - 0.168219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712587 - 0.168219i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
| 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
−15.85623487396185355848626453485, −15.05703307574073467618807206524, −13.60018089567225583580829239729, −12.63359234089730320646127387448, −11.53442586127015926715224259275, −10.16278367615633358640160989689, −8.113042478958499833782988951324, −7.25786572854953621558079942252, −5.69744188589088268818483882099, −2.78874555368698009720649366526,
3.98962169942933263048874949446, 5.11483059456094680659949847368, 7.50468587465601502535680789764, 9.026825353043187813555511512476, 10.14549722939175421713086739992, 11.31195885869802604868138270701, 12.79232493421510484778669632700, 14.04463007559186991357323450273, 15.52656668748200601127586916507, 16.19698357044329524928241643066
