| L(s) = 1 | − 3-s − 1.23i·5-s + 3.80·7-s + 9-s + (−3.23 − 0.726i)11-s + 2.35·13-s + 1.23i·15-s + 4.70i·17-s + 6.15i·19-s − 3.80·21-s + 3.23i·23-s + 3.47·25-s − 27-s − 1.45·29-s + 4.47i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.552i·5-s + 1.43·7-s + 0.333·9-s + (−0.975 − 0.219i)11-s + 0.652·13-s + 0.319i·15-s + 1.14i·17-s + 1.41i·19-s − 0.830·21-s + 0.674i·23-s + 0.694·25-s − 0.192·27-s − 0.269·29-s + 0.803i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.594769904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.594769904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (3.23 + 0.726i)T \) |
| good | 5 | \( 1 + 1.23iT - 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 4.70iT - 17T^{2} \) |
| 19 | \( 1 - 6.15iT - 19T^{2} \) |
| 23 | \( 1 - 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 - 4.47iT - 31T^{2} \) |
| 37 | \( 1 - 2.47iT - 37T^{2} \) |
| 41 | \( 1 + 7.60iT - 41T^{2} \) |
| 43 | \( 1 + 1.45iT - 43T^{2} \) |
| 47 | \( 1 - 0.763iT - 47T^{2} \) |
| 53 | \( 1 + 1.23iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 7.60iT - 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 9.06iT - 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.966927110435795595136884491429, −8.221473117512416319637634324392, −7.910107347994371770641980762690, −6.81278364768360591924933984125, −5.63856312045354208965959059334, −5.38441176776223817287482212543, −4.42524788697848444731639467702, −3.55808326859378872372549383487, −1.94879061938830197245101717653, −1.16013704122978724273032053679,
0.70650284639512759314809994900, 2.12397552767089335905003544555, 3.01346701929530867579331128017, 4.51366444619928753027747579356, 4.90110928886563776149062632297, 5.76931287980309730459112072912, 6.79318231687352017096279334081, 7.41719877479195978694776649199, 8.158605207537562780431187873210, 8.967010484808882620352251392681
