L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 6·11-s − 12-s + 13-s + 14-s + 16-s + 3·17-s + 18-s − 21-s + 6·22-s − 3·23-s − 24-s + 26-s − 27-s + 28-s − 3·29-s − 5·31-s + 32-s − 6·33-s + 3·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.218·21-s + 1.27·22-s − 0.625·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.898·31-s + 0.176·32-s − 1.04·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.47599835699897, −12.28502358040168, −11.86935712799822, −11.35921917445187, −11.10523947339208, −10.70212872708724, −9.961963805102304, −9.524630021875693, −9.311346798666675, −8.518995480605272, −8.065394125748281, −7.613594436003364, −7.023521662478595, −6.506756188725354, −6.303852759113252, −5.700324501140525, −5.254083112734658, −4.786980662134736, −4.092365962517221, −3.846780106851406, −3.416364762989964, −2.634247037738768, −1.887790113738519, −1.429907509579255, −0.9844153928236639, 0,
0.9844153928236639, 1.429907509579255, 1.887790113738519, 2.634247037738768, 3.416364762989964, 3.846780106851406, 4.092365962517221, 4.786980662134736, 5.254083112734658, 5.700324501140525, 6.303852759113252, 6.506756188725354, 7.023521662478595, 7.613594436003364, 8.065394125748281, 8.518995480605272, 9.311346798666675, 9.524630021875693, 9.961963805102304, 10.70212872708724, 11.10523947339208, 11.35921917445187, 11.86935712799822, 12.28502358040168, 12.47599835699897