| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 3·11-s + 14-s + 16-s − 3·17-s − 6·19-s − 20-s − 3·22-s − 2·23-s + 25-s + 28-s + 3·29-s + 3·31-s + 32-s − 3·34-s − 35-s − 37-s − 6·38-s − 40-s − 3·41-s − 43-s − 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.223·20-s − 0.639·22-s − 0.417·23-s + 1/5·25-s + 0.188·28-s + 0.557·29-s + 0.538·31-s + 0.176·32-s − 0.514·34-s − 0.169·35-s − 0.164·37-s − 0.973·38-s − 0.158·40-s − 0.468·41-s − 0.152·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.146451497996909742414741839962, −7.55947946358028496449425566831, −6.56405990058932844390013504676, −6.08056229568137165407273954903, −4.83332190202465636941309325541, −4.63919100492136925221940798677, −3.56154188220255317605428378165, −2.65455170884079945670816110578, −1.73957054101312727200784638700, 0,
1.73957054101312727200784638700, 2.65455170884079945670816110578, 3.56154188220255317605428378165, 4.63919100492136925221940798677, 4.83332190202465636941309325541, 6.08056229568137165407273954903, 6.56405990058932844390013504676, 7.55947946358028496449425566831, 8.146451497996909742414741839962
