| L(s) = 1 | − 442.·2-s + 6.47e4·4-s + 3.90e5·5-s + 2.47e7·7-s + 2.93e7·8-s − 1.72e8·10-s + 7.66e6·11-s − 3.40e9·13-s − 1.09e10·14-s − 2.14e10·16-s + 5.35e10·17-s − 1.29e11·19-s + 2.53e10·20-s − 3.39e9·22-s − 2.11e11·23-s + 1.52e11·25-s + 1.50e12·26-s + 1.60e12·28-s − 4.13e12·29-s − 5.31e12·31-s + 5.65e12·32-s − 2.36e13·34-s + 9.68e12·35-s + 1.85e13·37-s + 5.74e13·38-s + 1.14e13·40-s + 5.57e13·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.494·4-s + 0.447·5-s + 1.62·7-s + 0.618·8-s − 0.546·10-s + 0.0107·11-s − 1.15·13-s − 1.98·14-s − 1.24·16-s + 1.86·17-s − 1.75·19-s + 0.221·20-s − 0.0131·22-s − 0.562·23-s + 0.200·25-s + 1.41·26-s + 0.803·28-s − 1.53·29-s − 1.11·31-s + 0.909·32-s − 2.27·34-s + 0.727·35-s + 0.868·37-s + 2.14·38-s + 0.276·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 + 442.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.47e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 7.66e6T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.40e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 5.35e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.29e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.11e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.13e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.31e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.85e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.57e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.31e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.22e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.74e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 6.31e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.14e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.08e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.25e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 7.71e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.63e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.45e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.61e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 9.64e15T + 5.95e33T^{2} \) |
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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
−11.26976290627264239422468867938, −10.32402229981909508670238919969, −9.300489624105945855862619210107, −8.099790279875772905155931126125, −7.44614418398470677525399742786, −5.53809666913503281119241491292, −4.35585629913282994297234904028, −2.15213614070625841385683174133, −1.38119407959083327441366039144, 0,
1.38119407959083327441366039144, 2.15213614070625841385683174133, 4.35585629913282994297234904028, 5.53809666913503281119241491292, 7.44614418398470677525399742786, 8.099790279875772905155931126125, 9.300489624105945855862619210107, 10.32402229981909508670238919969, 11.26976290627264239422468867938
