L(s) = 1 | + (−0.752 + 1.19i)2-s + (−0.867 − 1.80i)4-s − 2.94i·5-s + 2.73i·7-s + (2.81 + 0.318i)8-s + (3.52 + 2.21i)10-s + 3.93·11-s − 4.74·13-s + (−3.27 − 2.05i)14-s + (−2.49 + 3.12i)16-s − 4.54i·17-s − i·19-s + (−5.31 + 2.55i)20-s + (−2.95 + 4.70i)22-s + 4.04·23-s + ⋯ |
L(s) = 1 | + (−0.532 + 0.846i)2-s + (−0.433 − 0.901i)4-s − 1.31i·5-s + 1.03i·7-s + (0.993 + 0.112i)8-s + (1.11 + 0.701i)10-s + 1.18·11-s − 1.31·13-s + (−0.874 − 0.549i)14-s + (−0.624 + 0.781i)16-s − 1.10i·17-s − 0.229i·19-s + (−1.18 + 0.571i)20-s + (−0.630 + 1.00i)22-s + 0.843·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975550 - 0.252664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975550 - 0.252664i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.752 - 1.19i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 + 2.94iT - 5T^{2} \) |
| 7 | \( 1 - 2.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.93T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 4.54iT - 17T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 + 7.49iT - 29T^{2} \) |
| 31 | \( 1 + 6.00iT - 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 - 1.77iT - 41T^{2} \) |
| 43 | \( 1 + 8.20iT - 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 + 7.86iT - 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 - 1.15T + 61T^{2} \) |
| 67 | \( 1 - 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 + 2.90iT - 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 3.45iT - 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.783768679171752702138278027628, −9.387318320539972422868472856972, −8.803796695783729205447997659054, −7.910234858855161297352602441597, −6.95867155429527904371649814316, −5.89597589073742755575840502254, −5.07211226201462570283945334191, −4.38161550680889328959162048939, −2.28087139405466293655715180746, −0.70857633111043794275256110601,
1.40907019154152344734011099302, 2.86460935240025730484726784081, 3.69342352416293103757519753493, 4.68750472542285686289460500516, 6.51329808434082762701797348499, 7.12674249692232526836754429232, 7.88363660398676633428749791840, 9.113901175880721218525665412032, 9.874838151101615909551443728816, 10.70771659042930853841768848585