| L(s) = 1 | − 2·3-s + 7-s + 9-s − 4·13-s − 2·19-s − 2·21-s − 5·25-s + 4·27-s + 6·29-s − 4·31-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s + 4·57-s + 6·59-s − 8·61-s + 63-s + 4·67-s − 2·73-s + 10·75-s + 8·79-s − 11·81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.458·19-s − 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s − 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.26272881518953, −14.89113096748579, −14.24967049524677, −13.72239902596623, −13.18434644610135, −12.30403017710645, −12.15788002311623, −11.70431473572361, −11.07013574289562, −10.56087889482214, −10.10510417175272, −9.570931276586979, −8.713971922191624, −8.345161402454639, −7.489142459129277, −7.078886987371320, −6.410367125615071, −5.858590903397252, −5.229737648703932, −4.845228980402497, −4.187234412870067, −3.380083826450902, −2.464917310380715, −1.838162648796834, −0.7894036805916308, 0,
0.7894036805916308, 1.838162648796834, 2.464917310380715, 3.380083826450902, 4.187234412870067, 4.845228980402497, 5.229737648703932, 5.858590903397252, 6.410367125615071, 7.078886987371320, 7.489142459129277, 8.345161402454639, 8.713971922191624, 9.570931276586979, 10.10510417175272, 10.56087889482214, 11.07013574289562, 11.70431473572361, 12.15788002311623, 12.30403017710645, 13.18434644610135, 13.72239902596623, 14.24967049524677, 14.89113096748579, 15.26272881518953
