L(s) = 1 | − 0.429·2-s − 1.81·4-s + 3.93·5-s + 0.264·7-s + 1.63·8-s − 1.68·10-s − 3.34·11-s + 0.652·13-s − 0.113·14-s + 2.92·16-s − 2.18·17-s + 4.72·19-s − 7.13·20-s + 1.43·22-s + 23-s + 10.4·25-s − 0.280·26-s − 0.480·28-s + 29-s − 2.95·31-s − 4.53·32-s + 0.940·34-s + 1.03·35-s − 10.8·37-s − 2.02·38-s + 6.44·40-s + 6.32·41-s + ⋯ |
L(s) = 1 | − 0.303·2-s − 0.907·4-s + 1.75·5-s + 0.0999·7-s + 0.579·8-s − 0.534·10-s − 1.00·11-s + 0.181·13-s − 0.0303·14-s + 0.731·16-s − 0.530·17-s + 1.08·19-s − 1.59·20-s + 0.306·22-s + 0.208·23-s + 2.09·25-s − 0.0550·26-s − 0.0907·28-s + 0.185·29-s − 0.530·31-s − 0.801·32-s + 0.161·34-s + 0.175·35-s − 1.77·37-s − 0.329·38-s + 1.01·40-s + 0.988·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.875413722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.875413722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 0.429T + 2T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 7 | \( 1 - 0.264T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 0.652T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 - 0.521T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 + 7.56T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 5.97T + 97T^{2} \) |
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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
−8.275942311751836203674350705174, −7.39545508425470242521512519779, −6.68383498302119006508573870563, −5.58599952517049369129848677065, −5.43417369644485923068799289819, −4.69769581229950905568794988769, −3.59796173497906908681434133194, −2.60259279990071966495304068876, −1.79758807088500697952401231412, −0.78024268530186785547632331366,
0.78024268530186785547632331366, 1.79758807088500697952401231412, 2.60259279990071966495304068876, 3.59796173497906908681434133194, 4.69769581229950905568794988769, 5.43417369644485923068799289819, 5.58599952517049369129848677065, 6.68383498302119006508573870563, 7.39545508425470242521512519779, 8.275942311751836203674350705174