| L(s) = 1 | + i·5-s − i·9-s + (−1 − i)13-s + (−1 + i)17-s − 25-s + (1 − i)37-s + 45-s + i·49-s + (1 + i)53-s + (1 − i)65-s + (1 + i)73-s − 81-s + (−1 − i)85-s + (1 − i)97-s − 2·101-s + ⋯ |
| L(s) = 1 | + i·5-s − i·9-s + (−1 − i)13-s + (−1 + i)17-s − 25-s + (1 − i)37-s + 45-s + i·49-s + (1 + i)53-s + (1 − i)65-s + (1 + i)73-s − 81-s + (−1 − i)85-s + (1 − i)97-s − 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6357565859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6357565859\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 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
−13.06560846434277772522137725342, −12.21353085036213570284905205661, −11.07925748798546145881732505980, −10.24071687327993332863047005091, −9.221074973098363193163920528040, −7.85863292749122721691212523124, −6.80059945319875688719833291978, −5.80140199949532992853905517256, −4.03087968401882776601714179342, −2.66147261116974021335730422679,
2.21888865321118726163530306768, 4.43966521530663007674469412826, 5.19020953810386242049906005307, 6.83939292138785200432351293434, 7.980117240652363769239370864423, 9.047836141025918656673687999356, 9.913649929430436329933245040040, 11.29170416241625925518280239708, 12.05397578143127107551365370655, 13.22226723029430935142271601052
