| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·11-s − 14-s + 16-s + 4·17-s − 19-s + 2·22-s + 3·23-s − 28-s − 2·29-s + 4·31-s + 32-s + 4·34-s − 10·37-s − 38-s − 7·41-s + 12·43-s + 2·44-s + 3·46-s − 8·47-s − 6·49-s − 9·53-s − 56-s − 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.426·22-s + 0.625·23-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.685·34-s − 1.64·37-s − 0.162·38-s − 1.09·41-s + 1.82·43-s + 0.301·44-s + 0.442·46-s − 1.16·47-s − 6/7·49-s − 1.23·53-s − 0.133·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.30911775475891, −13.79663347011373, −13.39003465399007, −12.76113533805331, −12.32275993859507, −12.03084484458998, −11.36119652318151, −10.89676014136648, −10.39681998326199, −9.714755557189016, −9.435820671336252, −8.673000370362778, −8.168029018223443, −7.569072064988548, −6.951335770807577, −6.539460339521378, −6.018849044447756, −5.375808594723829, −4.894849096918229, −4.290123876208060, −3.485660134276313, −3.310567757510899, −2.509470221672082, −1.686028057203916, −1.105766469531810, 0,
1.105766469531810, 1.686028057203916, 2.509470221672082, 3.310567757510899, 3.485660134276313, 4.290123876208060, 4.894849096918229, 5.375808594723829, 6.018849044447756, 6.539460339521378, 6.951335770807577, 7.569072064988548, 8.168029018223443, 8.673000370362778, 9.435820671336252, 9.714755557189016, 10.39681998326199, 10.89676014136648, 11.36119652318151, 12.03084484458998, 12.32275993859507, 12.76113533805331, 13.39003465399007, 13.79663347011373, 14.30911775475891
