L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s − 7-s + 0.999·8-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + 0.999·22-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s − 7-s + 0.999·8-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + 0.999·22-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3743785309\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3743785309\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.80836428889250303098667123006, −10.02319911829146260956479107459, −9.079886573931892356732382836531, −8.195791024364529130577245259835, −6.99044540008741661684377655260, −6.69650529050022840972059289102, −5.72079172578821552208646306030, −4.61746162742137428400038199163, −2.71756319310288640086755522984, −0.58359072686550270944538605702,
2.20929858287886171956204612219, 3.56463939227455072724241124916, 4.53342548396747871444269246282, 5.52278031065069754227797755913, 7.07674643995011102841156992952, 7.889587423899678767324600154886, 9.415898028367756444486774555359, 9.671652785089959592089114768578, 10.41918517941358334846528168972, 11.26246938350249775881440450465