L(s) = 1 | + 2.06·3-s + 5-s − 0.835·7-s + 1.28·9-s − 5.16·11-s + 4.52·13-s + 2.06·15-s − 5.30·17-s − 6.46·19-s − 1.72·21-s − 5.32·23-s + 25-s − 3.55·27-s + 7.21·29-s + 7.66·31-s − 10.6·33-s − 0.835·35-s + 7.65·37-s + 9.35·39-s − 3.14·41-s + 8.02·43-s + 1.28·45-s − 12.1·47-s − 6.30·49-s − 10.9·51-s + 11.3·53-s − 5.16·55-s + ⋯ |
L(s) = 1 | + 1.19·3-s + 0.447·5-s − 0.315·7-s + 0.427·9-s − 1.55·11-s + 1.25·13-s + 0.534·15-s − 1.28·17-s − 1.48·19-s − 0.377·21-s − 1.11·23-s + 0.200·25-s − 0.684·27-s + 1.33·29-s + 1.37·31-s − 1.86·33-s − 0.141·35-s + 1.25·37-s + 1.49·39-s − 0.490·41-s + 1.22·43-s + 0.191·45-s − 1.77·47-s − 0.900·49-s − 1.53·51-s + 1.55·53-s − 0.696·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 7 | \( 1 + 0.835T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 - 7.66T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 3.14T + 41T^{2} \) |
| 43 | \( 1 - 8.02T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 8.45T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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
−8.174050636052027209036941489860, −7.04113390672555096503888427705, −6.24846946040495195557480936079, −5.79867396874907828339826550652, −4.54269116173303139443885787660, −4.08996727175308259083524932827, −2.75477583079264952641076905782, −2.69691138167140702311420846131, −1.63274584579559044498753486014, 0,
1.63274584579559044498753486014, 2.69691138167140702311420846131, 2.75477583079264952641076905782, 4.08996727175308259083524932827, 4.54269116173303139443885787660, 5.79867396874907828339826550652, 6.24846946040495195557480936079, 7.04113390672555096503888427705, 8.174050636052027209036941489860