L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s + 3·8-s + 9-s + 2·10-s − 6.47·11-s + 12-s − 2·13-s + 2·15-s − 16-s + 4.47·17-s − 18-s − 6.47·19-s + 2·20-s + 6.47·22-s − 23-s − 3·24-s − 25-s + 2·26-s − 27-s + 29-s − 2·30-s − 8·31-s − 5·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.5·4-s − 0.894·5-s + 0.408·6-s + 1.06·8-s + 0.333·9-s + 0.632·10-s − 1.95·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s − 0.250·16-s + 1.08·17-s − 0.235·18-s − 1.48·19-s + 0.447·20-s + 1.37·22-s − 0.208·23-s − 0.612·24-s − 0.200·25-s + 0.392·26-s − 0.192·27-s + 0.185·29-s − 0.365·30-s − 1.43·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1708800024\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1708800024\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−9.097503522664397494269737288047, −8.250225379137463814134942452041, −7.69887750453685184898414395008, −7.23068243215044427121692976563, −5.84583734030000226310010687063, −5.08750031272530907332371021223, −4.40136335005202697510961132096, −3.37056710671189899513123215829, −1.97248292149484179888563619549, −0.30344002975376199490367420995,
0.30344002975376199490367420995, 1.97248292149484179888563619549, 3.37056710671189899513123215829, 4.40136335005202697510961132096, 5.08750031272530907332371021223, 5.84583734030000226310010687063, 7.23068243215044427121692976563, 7.69887750453685184898414395008, 8.250225379137463814134942452041, 9.097503522664397494269737288047