L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 3·17-s + 18-s + 7·19-s − 20-s − 2·21-s + 22-s − 4·23-s + 24-s − 4·25-s + 26-s + 27-s − 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 911 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.32028625774871, −13.96952594437785, −13.35775355524712, −12.97337341467782, −12.39323645572314, −12.01965936205811, −11.46404664367509, −11.01245444754360, −10.31145428739793, −9.749951756896713, −9.307880753096565, −8.964432506687106, −7.913199602710229, −7.546837915179151, −7.469916473873707, −6.445426000129157, −6.005286266652099, −5.552423698125427, −4.820513932821583, −4.020596577635311, −3.722255133785634, −3.197622742896585, −2.636299954049761, −1.771994594212414, −1.102190149934339, 0,
1.102190149934339, 1.771994594212414, 2.636299954049761, 3.197622742896585, 3.722255133785634, 4.020596577635311, 4.820513932821583, 5.552423698125427, 6.005286266652099, 6.445426000129157, 7.469916473873707, 7.546837915179151, 7.913199602710229, 8.964432506687106, 9.307880753096565, 9.749951756896713, 10.31145428739793, 11.01245444754360, 11.46404664367509, 12.01965936205811, 12.39323645572314, 12.97337341467782, 13.35775355524712, 13.96952594437785, 14.32028625774871