L(s) = 1 | − 0.102·2-s − 1.98·4-s − 2.62·5-s + 7-s + 0.408·8-s + 0.268·10-s + 0.541·11-s − 2.40·13-s − 0.102·14-s + 3.93·16-s − 1.67·17-s + 1.54·19-s + 5.22·20-s − 0.0553·22-s − 2.55·23-s + 1.90·25-s + 0.246·26-s − 1.98·28-s − 1.68·29-s − 1.59·31-s − 1.21·32-s + 0.171·34-s − 2.62·35-s + 2.71·37-s − 0.157·38-s − 1.07·40-s + 7.65·41-s + ⋯ |
L(s) = 1 | − 0.0723·2-s − 0.994·4-s − 1.17·5-s + 0.377·7-s + 0.144·8-s + 0.0849·10-s + 0.163·11-s − 0.667·13-s − 0.0273·14-s + 0.984·16-s − 0.406·17-s + 0.354·19-s + 1.16·20-s − 0.0118·22-s − 0.532·23-s + 0.380·25-s + 0.0483·26-s − 0.375·28-s − 0.312·29-s − 0.286·31-s − 0.215·32-s + 0.0293·34-s − 0.444·35-s + 0.445·37-s − 0.0256·38-s − 0.169·40-s + 1.19·41-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 | \( 1 + 0.102T + 2T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 11 | \( 1 - 0.541T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 7.24T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 0.0639T + 89T^{2} \) |
| 97 | \( 1 + 1.37T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57196157045397583524633216506, −7.11550211462489449025719149023, −5.95497691913601491193274912742, −5.29394353958893443700781312888, −4.42551962004078430197984265555, −4.12225933346474876232441269258, −3.32537697929350664456971916720, −2.25024216459053683301176991877, −0.955338175428465973200979205611, 0,
0.955338175428465973200979205611, 2.25024216459053683301176991877, 3.32537697929350664456971916720, 4.12225933346474876232441269258, 4.42551962004078430197984265555, 5.29394353958893443700781312888, 5.95497691913601491193274912742, 7.11550211462489449025719149023, 7.57196157045397583524633216506