L(s) = 1 | + 2·3-s − 2·4-s + 3·5-s + 7-s + 9-s − 4·12-s − 4·13-s + 6·15-s + 4·16-s + 3·17-s − 6·20-s + 2·21-s + 4·25-s − 4·27-s − 2·28-s + 6·29-s + 4·31-s + 3·35-s − 2·36-s − 2·37-s − 8·39-s − 6·41-s + 43-s + 3·45-s − 3·47-s + 8·48-s − 6·49-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 4-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.15·12-s − 1.10·13-s + 1.54·15-s + 16-s + 0.727·17-s − 1.34·20-s + 0.436·21-s + 4/5·25-s − 0.769·27-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.507·35-s − 1/3·36-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 0.152·43-s + 0.447·45-s − 0.437·47-s + 1.15·48-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 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 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.66316846801476, −14.33856682119376, −13.96109575948699, −13.54312919204235, −13.16507451758571, −12.39410251777930, −12.15802499208649, −11.24913908111154, −10.46818811105109, −9.922012831940696, −9.599236391319428, −9.367802799266532, −8.448858453902896, −8.285458397428332, −7.751643530904382, −6.912407972246272, −6.322973896061977, −5.453096261840122, −5.203115410959213, −4.541959481460336, −3.839053925215717, −2.972290989907395, −2.692989716240768, −1.791970373083437, −1.246228753992631, 0,
1.246228753992631, 1.791970373083437, 2.692989716240768, 2.972290989907395, 3.839053925215717, 4.541959481460336, 5.203115410959213, 5.453096261840122, 6.322973896061977, 6.912407972246272, 7.751643530904382, 8.285458397428332, 8.448858453902896, 9.367802799266532, 9.599236391319428, 9.922012831940696, 10.46818811105109, 11.24913908111154, 12.15802499208649, 12.39410251777930, 13.16507451758571, 13.54312919204235, 13.96109575948699, 14.33856682119376, 14.66316846801476