L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 4.18·5-s − 6·6-s + 3.18·7-s + 8·8-s + 9·9-s + 8.37·10-s + 69.4·11-s − 12·12-s − 8.12·13-s + 6.37·14-s − 12.5·15-s + 16·16-s − 106.·17-s + 18·18-s + 16.7·20-s − 9.56·21-s + 138.·22-s − 176.·23-s − 24·24-s − 107.·25-s − 16.2·26-s − 27·27-s + 12.7·28-s − 66.2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.374·5-s − 0.408·6-s + 0.172·7-s + 0.353·8-s + 0.333·9-s + 0.264·10-s + 1.90·11-s − 0.288·12-s − 0.173·13-s + 0.121·14-s − 0.216·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s + 0.187·20-s − 0.0993·21-s + 1.34·22-s − 1.60·23-s − 0.204·24-s − 0.859·25-s − 0.122·26-s − 0.192·27-s + 0.0860·28-s − 0.424·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 4.18T + 125T^{2} \) |
| 7 | \( 1 - 3.18T + 343T^{2} \) |
| 11 | \( 1 - 69.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.12T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 66.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 620.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 91.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 887.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 199.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 389.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 419.T + 9.12e5T^{2} \) |
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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
−8.330200960335170579360266432947, −7.20783576111380342442012195739, −6.54928791323552567030951085607, −6.02844590069196671982796596644, −5.15981203909107988450380637301, −4.14985879806247405964625764716, −3.76419964941232294033542249397, −2.16132234553381211054735264234, −1.53016274877462214274435150437, 0,
1.53016274877462214274435150437, 2.16132234553381211054735264234, 3.76419964941232294033542249397, 4.14985879806247405964625764716, 5.15981203909107988450380637301, 6.02844590069196671982796596644, 6.54928791323552567030951085607, 7.20783576111380342442012195739, 8.330200960335170579360266432947