| L(s) = 1 | − 1.77e5·3-s + 1.65e8·5-s − 2.22e9·7-s + 3.13e10·9-s − 1.31e12·11-s − 3.82e12·13-s − 2.92e13·15-s + 8.07e13·17-s − 8.71e14·19-s + 3.94e14·21-s + 1.70e15·23-s + 1.53e16·25-s − 5.55e15·27-s − 6.02e16·29-s + 1.11e17·31-s + 2.32e17·33-s − 3.67e17·35-s + 1.05e18·37-s + 6.77e17·39-s + 4.12e18·41-s + 2.84e18·43-s + 5.18e18·45-s + 2.95e19·47-s − 2.24e19·49-s − 1.43e19·51-s − 1.18e20·53-s − 2.16e20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·5-s − 0.425·7-s + 0.333·9-s − 1.38·11-s − 0.592·13-s − 0.872·15-s + 0.571·17-s − 1.71·19-s + 0.245·21-s + 0.373·23-s + 1.28·25-s − 0.192·27-s − 0.917·29-s + 0.788·31-s + 0.800·33-s − 0.643·35-s + 0.976·37-s + 0.341·39-s + 1.16·41-s + 0.466·43-s + 0.503·45-s + 1.74·47-s − 0.818·49-s − 0.330·51-s − 1.75·53-s − 2.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(1.678959392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678959392\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.77e5T \) |
| good | 5 | \( 1 - 1.65e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 2.22e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.31e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 3.82e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 8.07e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 8.71e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 1.70e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 6.02e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.11e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.05e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.12e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 2.84e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.95e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.18e20T + 4.55e39T^{2} \) |
| 59 | \( 1 + 8.18e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 3.33e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 9.91e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 5.40e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.68e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 5.65e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.86e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.19e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 8.60e22T + 4.96e45T^{2} \) |
| 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
−10.88807963955829151139791267301, −10.15824925362641257616832406420, −9.265166411004117197331633079766, −7.70496733401483334433345532684, −6.33301171535702881532223773601, −5.65049667263393215603723891924, −4.60267594827168814616430895132, −2.76704973202876496479389843675, −1.96160978113711826355208671156, −0.56309032170048449904397986996,
0.56309032170048449904397986996, 1.96160978113711826355208671156, 2.76704973202876496479389843675, 4.60267594827168814616430895132, 5.65049667263393215603723891924, 6.33301171535702881532223773601, 7.70496733401483334433345532684, 9.265166411004117197331633079766, 10.15824925362641257616832406420, 10.88807963955829151139791267301
