L(s) = 1 | + 3-s − 2·4-s + 2·7-s + 9-s + 11-s − 2·12-s − 6·13-s + 4·16-s − 4·17-s − 2·19-s + 2·21-s − 3·23-s + 27-s − 4·28-s + 29-s − 4·31-s + 33-s − 2·36-s + 3·37-s − 6·39-s + 7·41-s − 5·43-s − 2·44-s − 6·47-s + 4·48-s − 3·49-s − 4·51-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 1.66·13-s + 16-s − 0.970·17-s − 0.458·19-s + 0.436·21-s − 0.625·23-s + 0.192·27-s − 0.755·28-s + 0.185·29-s − 0.718·31-s + 0.174·33-s − 1/3·36-s + 0.493·37-s − 0.960·39-s + 1.09·41-s − 0.762·43-s − 0.301·44-s − 0.875·47-s + 0.577·48-s − 3/7·49-s − 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.616145362225899285091988408871, −8.061638128858403786474137913687, −7.36278754958494017460252627790, −6.37285416545749887240215303884, −5.17522095389622481613177994570, −4.61302926044132737393909951738, −3.92515695251871556359483309630, −2.69254658465815446366551411536, −1.68273659030778080859428751300, 0,
1.68273659030778080859428751300, 2.69254658465815446366551411536, 3.92515695251871556359483309630, 4.61302926044132737393909951738, 5.17522095389622481613177994570, 6.37285416545749887240215303884, 7.36278754958494017460252627790, 8.061638128858403786474137913687, 8.616145362225899285091988408871