| L(s) = 1 | + 0.697·5-s − 7-s − 5.60·11-s + 2.30·13-s + 0.394·17-s − 0.394·19-s + 23-s − 4.51·25-s + 5.60·29-s − 3.60·31-s − 0.697·35-s + 5.60·37-s − 3.60·41-s + 4.30·43-s + 4.60·47-s + 49-s − 5.90·53-s − 3.90·55-s − 3.90·59-s + 4.90·61-s + 1.60·65-s + 13.3·67-s − 12.9·71-s + 11·73-s + 5.60·77-s + 13.4·79-s − 16.2·83-s + ⋯ |
| L(s) = 1 | + 0.311·5-s − 0.377·7-s − 1.69·11-s + 0.638·13-s + 0.0956·17-s − 0.0904·19-s + 0.208·23-s − 0.902·25-s + 1.04·29-s − 0.647·31-s − 0.117·35-s + 0.921·37-s − 0.563·41-s + 0.656·43-s + 0.671·47-s + 0.142·49-s − 0.811·53-s − 0.526·55-s − 0.508·59-s + 0.628·61-s + 0.199·65-s + 1.62·67-s − 1.53·71-s + 1.28·73-s + 0.638·77-s + 1.51·79-s − 1.77·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577534265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577534265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.697T + 5T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 0.394T + 17T^{2} \) |
| 19 | \( 1 + 0.394T + 19T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 + 3.60T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 5.90T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 3.78T + 97T^{2} \) |
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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
−8.033742371992965238889687482786, −7.53898943339596191484096691629, −6.61642497375424275636717907633, −5.92092618127730957432540547140, −5.33272403564550635146535916860, −4.53526604508700414919015088113, −3.56285688019199142874781117842, −2.76597192447442870121819746210, −1.99475099803468945289144481238, −0.64522192327729755242591640609,
0.64522192327729755242591640609, 1.99475099803468945289144481238, 2.76597192447442870121819746210, 3.56285688019199142874781117842, 4.53526604508700414919015088113, 5.33272403564550635146535916860, 5.92092618127730957432540547140, 6.61642497375424275636717907633, 7.53898943339596191484096691629, 8.033742371992965238889687482786
