L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.123 − 0.0841i)3-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (−0.134 + 0.0648i)6-s + (0.955 + 0.294i)7-s + (−0.900 − 0.433i)8-s + (−0.357 + 0.910i)9-s + (0.0747 − 0.997i)10-s + (0.142 − 0.0440i)12-s + (−0.900 − 0.433i)14-s + (0.0931 + 0.116i)15-s + (0.826 + 0.563i)16-s + (0.488 − 0.846i)18-s + (−0.222 + 0.974i)20-s + (0.142 − 0.0440i)21-s + ⋯ |
L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.123 − 0.0841i)3-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (−0.134 + 0.0648i)6-s + (0.955 + 0.294i)7-s + (−0.900 − 0.433i)8-s + (−0.357 + 0.910i)9-s + (0.0747 − 0.997i)10-s + (0.142 − 0.0440i)12-s + (−0.900 − 0.433i)14-s + (0.0931 + 0.116i)15-s + (0.826 + 0.563i)16-s + (0.488 − 0.846i)18-s + (−0.222 + 0.974i)20-s + (0.142 − 0.0440i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7213519599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7213519599\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T \) |
| 7 | \( 1 + (-0.955 - 0.294i)T \) |
good | 3 | \( 1 + (-0.123 + 0.0841i)T + (0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 13 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-1.78 - 0.268i)T + (0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 61 | \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.535 - 1.36i)T + (-0.733 - 0.680i)T^{2} \) |
| 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
−10.28242364485217067125253520467, −9.661879989576771645539526115588, −8.505089089377443651543884060801, −7.907950049201490274392969679985, −7.33568834154131008234013561661, −6.22775077298781032201449211657, −5.38777050084086605081875364970, −3.85579234042117244733015360032, −2.56263658912123895284537744722, −1.88271787752673561190279379052,
0.971870497380337700571364278086, 2.20316738035554898954803748318, 3.75855523523910320958987785217, 4.93265450947127390646766418319, 5.86134770250902083395262956498, 6.82838782857256304986763482881, 7.83218872404889209816792305855, 8.637050342351906346952729371890, 8.904595219863915046174077118002, 10.01660491664782596458974011881