| L(s) = 1 | − 7-s + 2·11-s + 2·13-s + 6·17-s − 4·23-s − 6·29-s + 2·31-s + 2·37-s − 6·41-s + 2·43-s + 2·47-s + 49-s + 2·53-s + 4·61-s − 2·67-s + 8·71-s − 2·73-s − 2·77-s − 4·79-s − 12·83-s − 6·89-s − 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s + 0.554·13-s + 1.45·17-s − 0.834·23-s − 1.11·29-s + 0.359·31-s + 0.328·37-s − 0.937·41-s + 0.304·43-s + 0.291·47-s + 1/7·49-s + 0.274·53-s + 0.512·61-s − 0.244·67-s + 0.949·71-s − 0.234·73-s − 0.227·77-s − 0.450·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.355056493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.355056493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.81821315006278, −13.27220747498800, −12.74654966812468, −12.24052343901960, −11.86104518417159, −11.35487181283558, −10.80775440922054, −10.24727655359860, −9.662318367763730, −9.536271455106370, −8.687866813266099, −8.315942419023085, −7.760768118583736, −7.161781727404700, −6.732962066334796, −5.943795558972203, −5.769606640906803, −5.121789322224595, −4.278209788208339, −3.838772414206047, −3.337501613864957, −2.715469311762750, −1.853235079320592, −1.289110045265215, −0.5044599428085122,
0.5044599428085122, 1.289110045265215, 1.853235079320592, 2.715469311762750, 3.337501613864957, 3.838772414206047, 4.278209788208339, 5.121789322224595, 5.769606640906803, 5.943795558972203, 6.732962066334796, 7.161781727404700, 7.760768118583736, 8.315942419023085, 8.687866813266099, 9.536271455106370, 9.662318367763730, 10.24727655359860, 10.80775440922054, 11.35487181283558, 11.86104518417159, 12.24052343901960, 12.74654966812468, 13.27220747498800, 13.81821315006278
