| L(s) = 1 | − i·5-s + (−1 − i)7-s − 11-s + (−1 − i)17-s + i·19-s + (−1 + i)23-s − 25-s − i·29-s + 31-s + (−1 + i)35-s − 41-s + i·49-s + (1 − i)53-s + i·55-s − i·59-s + ⋯ |
| L(s) = 1 | − i·5-s + (−1 − i)7-s − 11-s + (−1 − i)17-s + i·19-s + (−1 + i)23-s − 25-s − i·29-s + 31-s + (−1 + i)35-s − 41-s + i·49-s + (1 − i)53-s + i·55-s − i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5558104945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5558104945\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.528469047834042388493293917643, −8.326325336335636022341127485430, −7.80050111670424135503191945327, −6.87441862660598577907019260160, −5.99615877641375413605742674760, −5.08060148370656358236704242125, −4.22266232440364836589613707871, −3.36450347605110785160769639157, −2.02801588623418918675538848601, −0.39798184820396681645332112451,
2.34158363970689025362885140343, 2.77690535333401520316094638848, 3.89968880573234909674772514469, 5.07487114245196319272723291505, 6.13953809572977233238603293969, 6.50752955969908720603522920448, 7.43077438352719075513547102893, 8.480971055099012828025218778166, 9.009293671166468499404791861215, 10.19737043187140351711801789303
