L(s) = 1 | + (−0.900 + 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)7-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)12-s + (−1.12 − 1.40i)13-s + (−0.900 − 0.433i)16-s + 1.24·19-s + (−0.900 − 0.433i)21-s + (0.623 − 0.781i)25-s + (−0.222 + 0.974i)27-s + (−0.900 + 0.433i)28-s − 0.445·31-s + (0.623 + 0.781i)36-s + (0.0990 + 0.433i)37-s + ⋯ |
L(s) = 1 | + (−0.900 + 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)7-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)12-s + (−1.12 − 1.40i)13-s + (−0.900 − 0.433i)16-s + 1.24·19-s + (−0.900 − 0.433i)21-s + (0.623 − 0.781i)25-s + (−0.222 + 0.974i)27-s + (−0.900 + 0.433i)28-s − 0.445·31-s + (0.623 + 0.781i)36-s + (0.0990 + 0.433i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5030964000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5030964000\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.900 - 0.433i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
good | 2 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + 1.80T + T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−13.14212526949402757005998691217, −12.19107042111006026078096654767, −11.76800986394345354496296409303, −10.50552222409029036954061636298, −9.421846044331542266949467387515, −8.216512192743882980429725440452, −7.18338939087353266449379690246, −5.54992912851620471209409676054, −4.70153546443400454209523227185, −3.03019981309096679580850642751,
1.58715089653729890030981512683, 4.52402876912039506229677954468, 5.32065799767493430308145670719, 6.72254909617020985484224982718, 7.50644733429563399858112453541, 9.283901789721509241191267639385, 10.22638645859426854822911819067, 11.20449055841784289390214617734, 11.88178611580306245556353757463, 13.25732861719228450539883026316