L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 12-s − 6·13-s − 14-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s + 21-s − 24-s + 25-s − 6·26-s − 27-s − 28-s − 6·29-s + 30-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.95801171153473, −13.22439990147096, −12.82662917635095, −12.31216344033439, −12.15943241854957, −11.64998702792955, −10.91471938296668, −10.71275785506588, −9.985616436900571, −9.621921918365169, −9.131199855012165, −8.267268775390272, −7.690217609503055, −7.306248001859313, −6.999793564712032, −6.114747298501623, −5.769403660342042, −5.343744503380888, −4.604957094573654, −4.195381671189611, −3.739358406510193, −2.749839792645738, −2.615310059610333, −1.635549843735448, −0.7918563834238269, 0,
0.7918563834238269, 1.635549843735448, 2.615310059610333, 2.749839792645738, 3.739358406510193, 4.195381671189611, 4.604957094573654, 5.343744503380888, 5.769403660342042, 6.114747298501623, 6.999793564712032, 7.306248001859313, 7.690217609503055, 8.267268775390272, 9.131199855012165, 9.621921918365169, 9.985616436900571, 10.71275785506588, 10.91471938296668, 11.64998702792955, 12.15943241854957, 12.31216344033439, 12.82662917635095, 13.22439990147096, 13.95801171153473