L(s) = 1 | − 2-s − 3-s + 4-s + 1.77·5-s + 6-s + 2.51·7-s − 8-s + 9-s − 1.77·10-s + 11-s − 12-s − 6.37·13-s − 2.51·14-s − 1.77·15-s + 16-s + 0.125·17-s − 18-s + 1.93·19-s + 1.77·20-s − 2.51·21-s − 22-s + 6.49·23-s + 24-s − 1.84·25-s + 6.37·26-s − 27-s + 2.51·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.794·5-s + 0.408·6-s + 0.952·7-s − 0.353·8-s + 0.333·9-s − 0.562·10-s + 0.301·11-s − 0.288·12-s − 1.76·13-s − 0.673·14-s − 0.458·15-s + 0.250·16-s + 0.0303·17-s − 0.235·18-s + 0.443·19-s + 0.397·20-s − 0.549·21-s − 0.213·22-s + 1.35·23-s + 0.204·24-s − 0.368·25-s + 1.24·26-s − 0.192·27-s + 0.476·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 - 0.125T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 0.558T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 0.0314T + 73T^{2} \) |
| 79 | \( 1 - 0.296T + 79T^{2} \) |
| 83 | \( 1 + 5.80T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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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
−7.924790271245431722083477127092, −7.36910089922798617034705482516, −6.82593151950317453940452756993, −5.73792188723115105388368734538, −5.25261397508225639117185155802, −4.54119660902399518806844073757, −3.21136820300652684212357880163, −2.06164566843275293820219087746, −1.48536966765541131427338374957, 0,
1.48536966765541131427338374957, 2.06164566843275293820219087746, 3.21136820300652684212357880163, 4.54119660902399518806844073757, 5.25261397508225639117185155802, 5.73792188723115105388368734538, 6.82593151950317453940452756993, 7.36910089922798617034705482516, 7.924790271245431722083477127092