L(s) = 1 | − 0.692·2-s − 1.52·4-s + 2.78·5-s + 7-s + 2.43·8-s − 1.92·10-s + 0.348·11-s + 2.17·13-s − 0.692·14-s + 1.35·16-s − 7.42·17-s − 0.255·19-s − 4.22·20-s − 0.241·22-s + 4.98·23-s + 2.73·25-s − 1.50·26-s − 1.52·28-s + 6.88·29-s − 10.1·31-s − 5.81·32-s + 5.14·34-s + 2.78·35-s − 1.64·37-s + 0.176·38-s + 6.77·40-s − 3.56·41-s + ⋯ |
L(s) = 1 | − 0.489·2-s − 0.760·4-s + 1.24·5-s + 0.377·7-s + 0.861·8-s − 0.608·10-s + 0.105·11-s + 0.603·13-s − 0.185·14-s + 0.338·16-s − 1.80·17-s − 0.0586·19-s − 0.945·20-s − 0.0514·22-s + 1.03·23-s + 0.547·25-s − 0.295·26-s − 0.287·28-s + 1.27·29-s − 1.81·31-s − 1.02·32-s + 0.881·34-s + 0.470·35-s − 0.270·37-s + 0.0287·38-s + 1.07·40-s − 0.557·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 0.692T + 2T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 11 | \( 1 - 0.348T + 11T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 17 | \( 1 + 7.42T + 17T^{2} \) |
| 19 | \( 1 + 0.255T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 - 6.88T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 + 7.69T + 61T^{2} \) |
| 67 | \( 1 + 6.52T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 + 1.68T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 0.216T + 97T^{2} \) |
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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
−7.54224270100350711982299852284, −6.75453793499332456623366081542, −6.18721138304051320835411031050, −5.26402697271315509570066111256, −4.82520928647352205416401258425, −4.02367688604606130640295443209, −3.00822341110264516281721200623, −1.89957056949398439560296119748, −1.38055266914188754748785379500, 0,
1.38055266914188754748785379500, 1.89957056949398439560296119748, 3.00822341110264516281721200623, 4.02367688604606130640295443209, 4.82520928647352205416401258425, 5.26402697271315509570066111256, 6.18721138304051320835411031050, 6.75453793499332456623366081542, 7.54224270100350711982299852284