L(s) = 1 | − i·9-s + (1 + i)13-s + (1 − i)17-s + (−1 + i)37-s + i·49-s + (−1 − i)53-s + (−1 − i)73-s − 81-s + (−1 + i)97-s − 2·101-s + 2i·109-s + (1 + i)113-s + (1 − i)117-s + ⋯ |
L(s) = 1 | − i·9-s + (1 + i)13-s + (1 − i)17-s + (−1 + i)37-s + i·49-s + (−1 − i)53-s + (−1 − i)73-s − 81-s + (−1 + i)97-s − 2·101-s + 2i·109-s + (1 + i)113-s + (1 − i)117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.017967173\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017967173\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−10.37508677141032781256638495775, −9.473674477475567967323353023605, −8.917100478281945147692390632369, −7.902183045738170981526358898506, −6.85352309348448355130523764019, −6.20513009170500104962279076153, −5.09689581356146083295485265421, −3.94475876647355079220750533822, −3.05444402249148542367272708883, −1.36895002068491677992741418241,
1.58849934508451499306883648860, 3.04004687598663522254644475155, 4.06483515617659980911558398079, 5.35050401038494667986764952731, 5.92165793282480784092446772764, 7.17792146549312341326590544182, 8.077543216265555129976011944201, 8.572212431423190890905315262018, 9.822517529218652472810256623789, 10.59992151531658934819817186097