L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 6·13-s + 15-s + 3·17-s − 4·19-s + 21-s + 6·23-s − 4·25-s + 27-s − 6·31-s + 35-s − 6·37-s − 6·39-s + 10·41-s + 11·43-s + 45-s − 9·47-s + 49-s + 3·51-s − 12·53-s − 4·57-s + 7·59-s + 2·61-s + 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.258·15-s + 0.727·17-s − 0.917·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 1.07·31-s + 0.169·35-s − 0.986·37-s − 0.960·39-s + 1.56·41-s + 1.67·43-s + 0.149·45-s − 1.31·47-s + 1/7·49-s + 0.420·51-s − 1.64·53-s − 0.529·57-s + 0.911·59-s + 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.01435718518768, −15.05943529902226, −14.74526051786881, −14.44678882665810, −13.86039850062736, −13.14397047319105, −12.62995850398760, −12.30221640038887, −11.50009178416421, −10.84841614273450, −10.35979631991251, −9.643201466219138, −9.294494828208968, −8.749038419805519, −7.810805292271469, −7.606594586785566, −6.929947649129181, −6.179222523779112, −5.408518163040914, −4.907579113474553, −4.224721361349761, −3.424448499097899, −2.607562688960250, −2.112554058847468, −1.269466602171861, 0,
1.269466602171861, 2.112554058847468, 2.607562688960250, 3.424448499097899, 4.224721361349761, 4.907579113474553, 5.408518163040914, 6.179222523779112, 6.929947649129181, 7.606594586785566, 7.810805292271469, 8.749038419805519, 9.294494828208968, 9.643201466219138, 10.35979631991251, 10.84841614273450, 11.50009178416421, 12.30221640038887, 12.62995850398760, 13.14397047319105, 13.86039850062736, 14.44678882665810, 14.74526051786881, 15.05943529902226, 16.01435718518768