L(s) = 1 | − 0.231i·2-s − 3.32·3-s + 1.94·4-s + 2.23i·5-s + 0.768i·6-s − i·7-s − 0.913i·8-s + 8.03·9-s + 0.516·10-s + 3.32i·11-s − 6.46·12-s − 0.231·14-s − 7.41i·15-s + 3.68·16-s + 1.37·17-s − 1.85i·18-s + ⋯ |
L(s) = 1 | − 0.163i·2-s − 1.91·3-s + 0.973·4-s + 0.997i·5-s + 0.313i·6-s − 0.377i·7-s − 0.322i·8-s + 2.67·9-s + 0.163·10-s + 1.00i·11-s − 1.86·12-s − 0.0618·14-s − 1.91i·15-s + 0.920·16-s + 0.333·17-s − 0.438i·18-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
\(L(1)\) |
\(\approx\) |
\(1.064164503\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.064164503\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.231iT - 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 - 2.23iT - 5T^{2} \) |
| 11 | \( 1 - 3.32iT - 11T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 3.23iT - 19T^{2} \) |
| 23 | \( 1 + 0.838T + 23T^{2} \) |
| 29 | \( 1 + 0.607T + 29T^{2} \) |
| 31 | \( 1 + 1.71iT - 31T^{2} \) |
| 37 | \( 1 - 1.55iT - 37T^{2} \) |
| 41 | \( 1 - 9.17iT - 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + 1.62iT - 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 - 8.82iT - 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 - 5.21iT - 71T^{2} \) |
| 73 | \( 1 + 3.96iT - 73T^{2} \) |
| 79 | \( 1 - 6.45T + 79T^{2} \) |
| 83 | \( 1 - 4.64iT - 83T^{2} \) |
| 89 | \( 1 - 9.12iT - 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−10.31167093434163032849650917489, −9.652245519636465125473269605903, −7.73413845839608585545245026945, −6.97718451559403144638552831303, −6.71252813142597171615164715306, −5.85236450901598381756347460780, −4.91825444010047641647545484934, −3.88349549384318340142674053311, −2.46557437504673626176629000262, −1.15174057452901347162689897001,
0.68851975095444636229562804839, 1.79193834449114359712912237937, 3.63482554156963940977545663370, 4.89726104643860147500911777199, 5.60223787372390552319039349980, 6.03361398760669114622706326985, 6.89760912430429960267016588950, 7.80885531878443427815425351200, 8.782330972081652586046860385206, 9.940249243371868044412791776439