L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 16.2·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 32.5·10-s + 29.7·11-s − 12·12-s − 46.5·13-s + 14·14-s + 48.8·15-s + 16·16-s − 12.5·17-s − 18·18-s + 19·19-s − 65.1·20-s + 21·21-s − 59.5·22-s + 113.·23-s + 24·24-s + 140.·25-s + 93.1·26-s − 27·27-s − 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.45·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.03·10-s + 0.816·11-s − 0.288·12-s − 0.993·13-s + 0.267·14-s + 0.841·15-s + 0.250·16-s − 0.178·17-s − 0.235·18-s + 0.229·19-s − 0.728·20-s + 0.218·21-s − 0.577·22-s + 1.02·23-s + 0.204·24-s + 1.12·25-s + 0.702·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 - 19T \) |
good | 5 | \( 1 + 16.2T + 125T^{2} \) |
| 11 | \( 1 - 29.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 97.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 41.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 26.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 465.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 52.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 32.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 568.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 376.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 452.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 222.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 765.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 105.T + 9.12e5T^{2} \) |
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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
−9.405694502414496120383874223338, −8.655318567592197616504934040307, −7.56287205090650517403840845193, −7.12689816361981885260474888891, −6.17882936044691037959103450225, −4.85283898169297330079317730664, −3.94358173359349410867211565202, −2.78758337654086005500480969957, −1.02446314920538743099175162483, 0,
1.02446314920538743099175162483, 2.78758337654086005500480969957, 3.94358173359349410867211565202, 4.85283898169297330079317730664, 6.17882936044691037959103450225, 7.12689816361981885260474888891, 7.56287205090650517403840845193, 8.655318567592197616504934040307, 9.405694502414496120383874223338