| L(s) = 1 | − 3-s − 1.57·5-s + 5.18·7-s + 9-s − 11-s + 13-s + 1.57·15-s − 4.76·17-s − 4.99·19-s − 5.18·21-s − 3.18·23-s − 2.51·25-s − 27-s − 4.23·29-s + 6.59·31-s + 33-s − 8.17·35-s + 10.3·37-s − 39-s + 7.18·41-s + 9.78·43-s − 1.57·45-s − 10.6·47-s + 19.8·49-s + 4.76·51-s − 0.329·53-s + 1.57·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.705·5-s + 1.95·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.407·15-s − 1.15·17-s − 1.14·19-s − 1.13·21-s − 0.663·23-s − 0.502·25-s − 0.192·27-s − 0.786·29-s + 1.18·31-s + 0.174·33-s − 1.38·35-s + 1.70·37-s − 0.160·39-s + 1.12·41-s + 1.49·43-s − 0.235·45-s − 1.56·47-s + 2.83·49-s + 0.666·51-s − 0.0452·53-s + 0.212·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.512014655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512014655\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 1.57T + 5T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.329T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 + 7.02T + 61T^{2} \) |
| 67 | \( 1 + 3.57T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 9.51T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 3.15T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.88808313850205196327506018807, −7.53905996808253652885585531820, −6.45896213373159278602045317372, −5.87666094604813254554235437330, −4.97038638549595738348961050664, −4.31475126239089763253263674693, −4.07645407822495013828101879461, −2.48190787809791536220189972926, −1.80431306609373019344988553028, −0.65290911447088810019641772735,
0.65290911447088810019641772735, 1.80431306609373019344988553028, 2.48190787809791536220189972926, 4.07645407822495013828101879461, 4.31475126239089763253263674693, 4.97038638549595738348961050664, 5.87666094604813254554235437330, 6.45896213373159278602045317372, 7.53905996808253652885585531820, 7.88808313850205196327506018807
