L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s − 2·11-s − 2·14-s + 16-s − 5·17-s − 2·19-s + 20-s − 2·22-s − 6·23-s − 4·25-s − 2·28-s + 9·29-s − 4·31-s + 32-s − 5·34-s − 2·35-s − 11·37-s − 2·38-s + 40-s − 5·41-s + 10·43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s − 0.603·11-s − 0.534·14-s + 1/4·16-s − 1.21·17-s − 0.458·19-s + 0.223·20-s − 0.426·22-s − 1.25·23-s − 4/5·25-s − 0.377·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.857·34-s − 0.338·35-s − 1.80·37-s − 0.324·38-s + 0.158·40-s − 0.780·41-s + 1.52·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.360681108862332625165692998100, −7.41874059760625422157718817115, −6.58662918477531782094318555426, −6.11162875978096597726272660139, −5.28104666371666603720689375117, −4.42024992364196241252116340582, −3.60495980956481521328674934972, −2.61422217324754074009844849983, −1.86372827665687110681755730342, 0,
1.86372827665687110681755730342, 2.61422217324754074009844849983, 3.60495980956481521328674934972, 4.42024992364196241252116340582, 5.28104666371666603720689375117, 6.11162875978096597726272660139, 6.58662918477531782094318555426, 7.41874059760625422157718817115, 8.360681108862332625165692998100