| L(s) = 1 | + 0.680·7-s + 1.68·11-s + 5.14·13-s + 1.31·17-s − 0.324·19-s − 3.78·23-s + 8.64·29-s − 4.14·31-s + 1.35·37-s + 7.15·41-s − 7.29·43-s + 12.9·47-s − 6.53·49-s + 8.83·53-s − 8.81·59-s + 9.97·61-s − 4.17·67-s + 0.891·71-s − 7.82·73-s + 1.14·77-s + 9.65·79-s − 4.85·83-s − 17.4·89-s + 3.50·91-s + 8.93·97-s + 10.8·101-s + 11.6·103-s + ⋯ |
| L(s) = 1 | + 0.257·7-s + 0.506·11-s + 1.42·13-s + 0.320·17-s − 0.0744·19-s − 0.790·23-s + 1.60·29-s − 0.744·31-s + 0.222·37-s + 1.11·41-s − 1.11·43-s + 1.89·47-s − 0.933·49-s + 1.21·53-s − 1.14·59-s + 1.27·61-s − 0.510·67-s + 0.105·71-s − 0.915·73-s + 0.130·77-s + 1.08·79-s − 0.533·83-s − 1.85·89-s + 0.366·91-s + 0.907·97-s + 1.08·101-s + 1.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.492821544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492821544\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 0.680T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 5.14T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 0.324T + 19T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 - 8.64T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 - 7.15T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 8.83T + 53T^{2} \) |
| 59 | \( 1 + 8.81T + 59T^{2} \) |
| 61 | \( 1 - 9.97T + 61T^{2} \) |
| 67 | \( 1 + 4.17T + 67T^{2} \) |
| 71 | \( 1 - 0.891T + 71T^{2} \) |
| 73 | \( 1 + 7.82T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 8.93T + 97T^{2} \) |
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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
−7.84420600209503557597001032965, −7.16759063610177248055472983196, −6.27035850052387872225835001427, −5.94982309025338254871825677719, −5.01799916430367604576739771383, −4.16170767875287173069654447859, −3.63877401264921874784845120663, −2.68497912062197213163286355143, −1.65563399792915422792173376191, −0.831177209002191274367505588560,
0.831177209002191274367505588560, 1.65563399792915422792173376191, 2.68497912062197213163286355143, 3.63877401264921874784845120663, 4.16170767875287173069654447859, 5.01799916430367604576739771383, 5.94982309025338254871825677719, 6.27035850052387872225835001427, 7.16759063610177248055472983196, 7.84420600209503557597001032965
