L(s) = 1 | + 2.73i·3-s − 0.732·5-s + (2 − 1.73i)7-s − 4.46·9-s + 5.46·11-s + 4.73·13-s − 2i·15-s + 4i·17-s + 1.26i·19-s + (4.73 + 5.46i)21-s − 5.46i·23-s − 4.46·25-s − 3.99i·27-s + 6.92i·29-s + 6.92·31-s + ⋯ |
L(s) = 1 | + 1.57i·3-s − 0.327·5-s + (0.755 − 0.654i)7-s − 1.48·9-s + 1.64·11-s + 1.31·13-s − 0.516i·15-s + 0.970i·17-s + 0.290i·19-s + (1.03 + 1.19i)21-s − 1.13i·23-s − 0.892·25-s − 0.769i·27-s + 1.28i·29-s + 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0716 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0716 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31553 + 1.22447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31553 + 1.22447i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 + 5.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2.92iT - 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6.92iT - 53T^{2} \) |
| 59 | \( 1 + 3.80iT - 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 4.53iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 - 14.9iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.46226675561690216441472918011, −9.519341304925639296462103127963, −8.682339309070620657472392701619, −8.175200179978594280155923945282, −6.75082922730444442436704367243, −5.89393277229525051921816937659, −4.64608611656547860784314597478, −4.02016285452699357988155122382, −3.50155169174521143539426026060, −1.41789064091893615861703309139,
1.07603577041752416904336029184, 1.91856583621132483853471710087, 3.30848256172512858456214678087, 4.58427785639446960445406731621, 5.98146381498554191637788670242, 6.41324342318245343264630003546, 7.42965418339472495126761822824, 8.132151101461361238436766100901, 8.808766164420323817614450525083, 9.699593610222403287632702125487