L(s) = 1 | + i·2-s + 1.02i·3-s − 4-s + 1.25i·5-s − 1.02·6-s − i·8-s + 1.94·9-s − 1.25·10-s + (−3.30 + 0.235i)11-s − 1.02i·12-s − 4.08·13-s − 1.29·15-s + 16-s + 3.20·17-s + 1.94i·18-s − 7.62·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.593i·3-s − 0.5·4-s + 0.562i·5-s − 0.419·6-s − 0.353i·8-s + 0.648·9-s − 0.398·10-s + (−0.997 + 0.0710i)11-s − 0.296i·12-s − 1.13·13-s − 0.333·15-s + 0.250·16-s + 0.776·17-s + 0.458i·18-s − 1.75·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5273867414\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5273867414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.30 - 0.235i)T \) |
good | 3 | \( 1 - 1.02iT - 3T^{2} \) |
| 5 | \( 1 - 1.25iT - 5T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 23 | \( 1 + 8.25T + 23T^{2} \) |
| 29 | \( 1 + 3.54iT - 29T^{2} \) |
| 31 | \( 1 - 9.18iT - 31T^{2} \) |
| 37 | \( 1 + 0.308T + 37T^{2} \) |
| 41 | \( 1 + 6.05T + 41T^{2} \) |
| 43 | \( 1 + 7.57iT - 43T^{2} \) |
| 47 | \( 1 - 4.70iT - 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 + 2.73iT - 59T^{2} \) |
| 61 | \( 1 - 1.51T + 61T^{2} \) |
| 67 | \( 1 - 3.38T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 + 0.966T + 73T^{2} \) |
| 79 | \( 1 + 15.6iT - 79T^{2} \) |
| 83 | \( 1 + 1.32T + 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−10.26062648750677995535156840803, −9.794098828284109570425975858145, −8.644536595190985385259409788818, −7.84810737338337143717547186382, −7.09753804399678661603591982197, −6.28640030371706037069287304514, −5.18396100354503841900146568379, −4.51835906224992779321465535840, −3.46722121762115789624397343537, −2.17162685400863230394580461587,
0.22106118038638732114123523929, 1.75234228523780288016581684567, 2.59700566619013517787608303563, 4.06189692571899793779450623741, 4.82411704432269375763360393901, 5.82236834511864268415957197788, 6.92591826364166055981427606507, 7.943842705070746625562601190220, 8.337175837617050268773895299673, 9.649011844583519663481083042800