L(s) = 1 | + (1.37 + 0.340i)2-s + (1.76 + 0.934i)4-s + (−2.75 + 2.75i)5-s + 0.113i·7-s + (2.10 + 1.88i)8-s + (−4.71 + 2.84i)10-s + (−3.69 + 3.69i)11-s + (−1.53 − 1.53i)13-s + (−0.0386 + 0.155i)14-s + (2.25 + 3.30i)16-s + 6.62·17-s + (3.27 + 3.27i)19-s + (−7.43 + 2.29i)20-s + (−6.32 + 3.81i)22-s − 6.20i·23-s + ⋯ |
L(s) = 1 | + (0.970 + 0.240i)2-s + (0.884 + 0.467i)4-s + (−1.23 + 1.23i)5-s + 0.0428i·7-s + (0.745 + 0.666i)8-s + (−1.49 + 0.898i)10-s + (−1.11 + 1.11i)11-s + (−0.426 − 0.426i)13-s + (−0.0103 + 0.0416i)14-s + (0.563 + 0.826i)16-s + 1.60·17-s + (0.751 + 0.751i)19-s + (−1.66 + 0.512i)20-s + (−1.34 + 0.812i)22-s − 1.29i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21226 + 1.49090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21226 + 1.49090i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.340i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.75 - 2.75i)T - 5iT^{2} \) |
| 7 | \( 1 - 0.113iT - 7T^{2} \) |
| 11 | \( 1 + (3.69 - 3.69i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.53 + 1.53i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 19 | \( 1 + (-3.27 - 3.27i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.20iT - 23T^{2} \) |
| 29 | \( 1 + (-1.51 - 1.51i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 + (-2.61 + 2.61i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.18iT - 41T^{2} \) |
| 43 | \( 1 + (-3.18 + 3.18i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 + (-4.08 + 4.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.942 + 0.942i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.437 + 0.437i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.26 + 3.26i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.72iT - 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + (8.42 + 8.42i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.361iT - 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−11.66898495594549112304638301144, −10.53810075391039325245588976388, −10.12827520276198175319080622017, −7.989968588887085286335781767358, −7.64960680219577839825276622477, −6.87045548134411850164782234182, −5.62066604519784882959954525794, −4.54843380097840203385210290280, −3.40651862165833586220761053069, −2.59829679394082883619295069005,
0.942562973207188617340066126406, 3.01909540993960456970441649599, 3.96970074552310840526207984713, 5.11620086351860435309777868915, 5.63262082123175057333636702196, 7.39785012221771984416000209249, 7.86420470772697073933530457300, 9.057081981148840070691499764088, 10.20605524472574194626207074990, 11.34848539812951528017607264628