L(s) = 1 | − 2·3-s + 2·4-s − 3i·5-s − i·7-s + 9-s − 4·12-s + 6i·15-s + 4·16-s + 6·17-s − 7i·19-s − 6i·20-s + 2i·21-s − 3·23-s − 4·25-s + 4·27-s − 2i·28-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 4-s − 1.34i·5-s − 0.377i·7-s + 0.333·9-s − 1.15·12-s + 1.54i·15-s + 16-s + 1.45·17-s − 1.60i·19-s − 1.34i·20-s + 0.436i·21-s − 0.625·23-s − 0.800·25-s + 0.769·27-s − 0.377i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.137792908\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.137792908\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 + 15iT - 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−9.575889413916126804594831808033, −8.677341569576823091609854290186, −7.65590970416138626610152838860, −6.97263834682839771865531714548, −5.89745373132924580817223792627, −5.40272482687157513786812509579, −4.55724141606123843471714834302, −3.23901684986306267091865053510, −1.65555022703813176395529685143, −0.56248222403482296849022218812,
1.63238860924544709848480149261, 2.87052458845859655850716669971, 3.71299663737923261566887746202, 5.38938690379834626450161030837, 6.05029007211674608258170698378, 6.40517329576887599962897624991, 7.54592372411120679323584788208, 7.943809659807535129151592911537, 9.685071767104213573392884697364, 10.24277281295338639514036251995