| L(s) = 1 | − 5-s − 5·11-s − 5·13-s − 17-s + 19-s + 3·23-s − 4·25-s − 2·29-s + 2·31-s − 8·37-s + 5·41-s − 9·43-s − 6·47-s − 7·49-s + 6·53-s + 5·55-s − 6·59-s − 4·61-s + 5·65-s + 12·67-s + 12·71-s − 2·73-s + 10·79-s + 2·83-s + 85-s − 12·89-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s − 1.38·13-s − 0.242·17-s + 0.229·19-s + 0.625·23-s − 4/5·25-s − 0.371·29-s + 0.359·31-s − 1.31·37-s + 0.780·41-s − 1.37·43-s − 0.875·47-s − 49-s + 0.824·53-s + 0.674·55-s − 0.781·59-s − 0.512·61-s + 0.620·65-s + 1.46·67-s + 1.42·71-s − 0.234·73-s + 1.12·79-s + 0.219·83-s + 0.108·85-s − 1.27·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−10.18490963672513276838211915410, −9.469029240875226710024119915125, −8.242710873880411872797331552948, −7.63736087168536861434605466701, −6.76959891375384935472749260030, −5.37641192809255730852979399474, −4.75477574120717256848740322395, −3.34157120180978815093157505643, −2.23127545325450121489119586078, 0,
2.23127545325450121489119586078, 3.34157120180978815093157505643, 4.75477574120717256848740322395, 5.37641192809255730852979399474, 6.76959891375384935472749260030, 7.63736087168536861434605466701, 8.242710873880411872797331552948, 9.469029240875226710024119915125, 10.18490963672513276838211915410
