L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 2·11-s − 12-s − 4·13-s − 14-s + 16-s − 18-s − 21-s − 2·22-s − 2·23-s + 24-s + 4·26-s − 27-s + 28-s − 6·29-s + 4·31-s − 32-s − 2·33-s + 36-s − 6·37-s + 4·39-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 + 0.603·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.218·21-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s − 0.986·37-s + 0.640·39-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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.47949695485988, −12.11307293459655, −11.82288987121719, −11.32762611813708, −10.81180441577398, −10.54908417975276, −9.943929237409724, −9.509393164659061, −9.238616536198865, −8.648659110784786, −8.107053270170822, −7.605512899193334, −7.310323487255161, −6.790761318662211, −6.244013063215127, −5.876042416893077, −5.239153131044226, −4.820473837742193, −4.282191589613555, −3.703180656522539, −3.107306523047373, −2.351135313826380, −1.945477295272368, −1.333195729467762, −0.6539263133812047, 0,
0.6539263133812047, 1.333195729467762, 1.945477295272368, 2.351135313826380, 3.107306523047373, 3.703180656522539, 4.282191589613555, 4.820473837742193, 5.239153131044226, 5.876042416893077, 6.244013063215127, 6.790761318662211, 7.310323487255161, 7.605512899193334, 8.107053270170822, 8.648659110784786, 9.238616536198865, 9.509393164659061, 9.943929237409724, 10.54908417975276, 10.81180441577398, 11.32762611813708, 11.82288987121719, 12.11307293459655, 12.47949695485988