L(s) = 1 | + (0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.326 − 0.118i)5-s + (0.173 − 0.984i)9-s + (−0.939 + 0.342i)11-s + (0.766 + 0.642i)12-s + (−0.326 + 0.118i)15-s + (−0.939 + 0.342i)16-s + (0.0603 − 0.342i)20-s + (−0.173 − 0.984i)23-s + (−0.673 − 0.565i)25-s + (−0.500 − 0.866i)27-s + (0.266 + 1.50i)31-s + (−0.5 + 0.866i)33-s + 36-s + (−0.173 − 0.300i)37-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.326 − 0.118i)5-s + (0.173 − 0.984i)9-s + (−0.939 + 0.342i)11-s + (0.766 + 0.642i)12-s + (−0.326 + 0.118i)15-s + (−0.939 + 0.342i)16-s + (0.0603 − 0.342i)20-s + (−0.173 − 0.984i)23-s + (−0.673 − 0.565i)25-s + (−0.500 − 0.866i)27-s + (0.266 + 1.50i)31-s + (−0.5 + 0.866i)33-s + 36-s + (−0.173 − 0.300i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9080947948\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9080947948\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.939 - 0.342i)T \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 37 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \) |
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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
−12.40928949119541438896602245903, −11.24431173327927600006780412288, −10.01923088151719031952640022397, −8.773741081934350896351335190754, −8.101897095820321120255472863980, −7.39431282861877647559838915823, −6.40881090642502171409726828797, −4.61173491729997113317236599889, −3.36417730849738865534218497816, −2.29619675943427701773647904468,
2.19370799218050459251125349540, 3.59681502528997047131938143207, 4.92723638224117684674520181114, 5.84471138623774864204276039501, 7.33529029505524823313294128049, 8.218698807691377846633469704155, 9.379752967256463840250575735044, 10.06024098828169117661847420476, 10.89710830963381586137116127701, 11.68454496604138985197082128615