| L(s) = 1 | − i·2-s + 0.445·3-s − 4-s + i·5-s − 0.445i·6-s + 3.24i·7-s + i·8-s − 2.80·9-s + 10-s − 1.60i·11-s − 0.445·12-s + 3.24·14-s + 0.445i·15-s + 16-s − 3.10·17-s + 2.80i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.256·3-s − 0.5·4-s + 0.447i·5-s − 0.181i·6-s + 1.22i·7-s + 0.353i·8-s − 0.933·9-s + 0.316·10-s − 0.483i·11-s − 0.128·12-s + 0.867·14-s + 0.114i·15-s + 0.250·16-s − 0.754·17-s + 0.660i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2931054354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2931054354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.445T + 3T^{2} \) |
| 7 | \( 1 - 3.24iT - 7T^{2} \) |
| 11 | \( 1 + 1.60iT - 11T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 19 | \( 1 + 2.89iT - 19T^{2} \) |
| 23 | \( 1 + 6.78T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 + 8.31iT - 31T^{2} \) |
| 37 | \( 1 + 3.20iT - 37T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 - 3.97iT - 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 5.28iT - 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 12.3iT - 67T^{2} \) |
| 71 | \( 1 + 7.50iT - 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 6.05iT - 97T^{2} \) |
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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
−9.090152107249930159482059246123, −8.375143000682262358332462660966, −7.62917094904280169254442835309, −6.18349725390519589682764457241, −5.84902546377622210531582869879, −4.70633106533432981004301956156, −3.61332617184372429156476922255, −2.63693773364641435899496283515, −2.15576736476948090354762430980, −0.10342981475313862934150192432,
1.47678397867495301607334778007, 3.02472136363226426617508838911, 4.13310067664699463821730732674, 4.71489576771687878618657847888, 5.81273382370289109373828120160, 6.57520667696885297814178823278, 7.41449505941970108106747184030, 8.190068436850264670659186561357, 8.659853152645122239509153983093, 9.714547162236281875617842142831
