| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s + 4·11-s + 13-s − 14-s − 16-s + 17-s − 6·19-s + 4·22-s + 26-s + 28-s − 7·31-s + 5·32-s + 34-s + 4·37-s − 6·38-s + 2·41-s − 4·43-s − 4·44-s − 6·47-s − 6·49-s − 52-s + 11·53-s + 3·56-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.20·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 0.242·17-s − 1.37·19-s + 0.852·22-s + 0.196·26-s + 0.188·28-s − 1.25·31-s + 0.883·32-s + 0.171·34-s + 0.657·37-s − 0.973·38-s + 0.312·41-s − 0.609·43-s − 0.603·44-s − 0.875·47-s − 6/7·49-s − 0.138·52-s + 1.51·53-s + 0.400·56-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.320416794306105352143360065819, −7.21357055910767856019499423548, −6.39378148855915660318852677761, −5.94524391459660288016771777062, −5.02314892599740851324178400191, −4.13590146952641728531203099741, −3.73535955610611468017822916783, −2.75191142520846876125282797421, −1.46609936691492573608798108200, 0,
1.46609936691492573608798108200, 2.75191142520846876125282797421, 3.73535955610611468017822916783, 4.13590146952641728531203099741, 5.02314892599740851324178400191, 5.94524391459660288016771777062, 6.39378148855915660318852677761, 7.21357055910767856019499423548, 8.320416794306105352143360065819
