| L(s) = 1 | − 8.27·2-s + 36.4·4-s − 28.7·5-s + 49·7-s − 37.0·8-s + 237.·10-s + 270.·11-s + 300.·13-s − 405.·14-s − 860.·16-s − 613.·17-s − 1.70e3·19-s − 1.04e3·20-s − 2.23e3·22-s − 3.18e3·23-s − 2.29e3·25-s − 2.48e3·26-s + 1.78e3·28-s − 4.29e3·29-s + 2.02e3·31-s + 8.30e3·32-s + 5.07e3·34-s − 1.40e3·35-s + 5.15e3·37-s + 1.40e4·38-s + 1.06e3·40-s + 7.14e3·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.13·4-s − 0.514·5-s + 0.377·7-s − 0.204·8-s + 0.752·10-s + 0.673·11-s + 0.493·13-s − 0.552·14-s − 0.840·16-s − 0.514·17-s − 1.08·19-s − 0.586·20-s − 0.984·22-s − 1.25·23-s − 0.735·25-s − 0.721·26-s + 0.430·28-s − 0.949·29-s + 0.379·31-s + 1.43·32-s + 0.752·34-s − 0.194·35-s + 0.618·37-s + 1.58·38-s + 0.105·40-s + 0.663·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 + 8.27T + 32T^{2} \) |
| 5 | \( 1 + 28.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 270.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 300.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 613.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.94e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.05e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.36e4T + 8.58e9T^{2} \) |
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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
−13.40068732379135880809024521843, −11.79235286957854930258046095508, −10.96738464651369409102198206617, −9.770019056205913469553285644602, −8.622324041671088209687833388445, −7.80627524892866774731121675700, −6.41975590104036289706913594785, −4.18210231475701305866080517294, −1.76249649689538642059159317241, 0,
1.76249649689538642059159317241, 4.18210231475701305866080517294, 6.41975590104036289706913594785, 7.80627524892866774731121675700, 8.622324041671088209687833388445, 9.770019056205913469553285644602, 10.96738464651369409102198206617, 11.79235286957854930258046095508, 13.40068732379135880809024521843
