L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s + 3·13-s − 15-s − 7·19-s − 21-s − 6·23-s − 4·25-s − 27-s − 9·29-s − 33-s + 35-s − 3·37-s − 3·39-s + 8·41-s − 10·43-s + 45-s − 3·47-s + 49-s + 6·53-s + 55-s + 7·57-s − 7·59-s + 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s − 0.258·15-s − 1.60·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.174·33-s + 0.169·35-s − 0.493·37-s − 0.480·39-s + 1.24·41-s − 1.52·43-s + 0.149·45-s − 0.437·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.927·57-s − 0.911·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.228825260092953460774761093275, −7.38085313284840354184236895709, −6.45929621338601453055359408770, −5.97257822119111493201281408238, −5.29985992796315974464476621233, −4.22014639315065978093340000706, −3.73508121125190193462431878713, −2.21360115742485206155461583084, −1.54436114815519325492532681257, 0,
1.54436114815519325492532681257, 2.21360115742485206155461583084, 3.73508121125190193462431878713, 4.22014639315065978093340000706, 5.29985992796315974464476621233, 5.97257822119111493201281408238, 6.45929621338601453055359408770, 7.38085313284840354184236895709, 8.228825260092953460774761093275