| L(s) = 1 | + 1.80i·2-s − 1.55·3-s − 1.24·4-s − i·5-s − 2.80i·6-s − 1.55i·7-s + 1.35i·8-s − 0.582·9-s + 1.80·10-s + 0.356i·11-s + 1.93·12-s + 2.80·14-s + 1.55i·15-s − 4.93·16-s + 1.33·17-s − 1.04i·18-s + ⋯ |
| L(s) = 1 | + 1.27i·2-s − 0.897·3-s − 0.623·4-s − 0.447i·5-s − 1.14i·6-s − 0.587i·7-s + 0.479i·8-s − 0.194·9-s + 0.569·10-s + 0.107i·11-s + 0.559·12-s + 0.748·14-s + 0.401i·15-s − 1.23·16-s + 0.323·17-s − 0.247i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00010 + 0.401682i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00010 + 0.401682i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.80iT - 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 7 | \( 1 + 1.55iT - 7T^{2} \) |
| 11 | \( 1 - 0.356iT - 11T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 + 8.65iT - 19T^{2} \) |
| 23 | \( 1 - 8.00T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 5.43iT - 31T^{2} \) |
| 37 | \( 1 - 4.60iT - 37T^{2} \) |
| 41 | \( 1 + 8.58iT - 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + 3.41iT - 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 + 0.506iT - 59T^{2} \) |
| 61 | \( 1 + 7.24T + 61T^{2} \) |
| 67 | \( 1 - 4.48iT - 67T^{2} \) |
| 71 | \( 1 - 6.39iT - 71T^{2} \) |
| 73 | \( 1 + 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 - 5.48iT - 83T^{2} \) |
| 89 | \( 1 + 16.5iT - 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−10.45456989858545129536794100557, −9.027817062076469532245178908395, −8.623511889079784605668986595525, −7.34027076087581707852706957321, −6.89739652309002254757911451859, −6.02595261152573331324152562156, −5.04216256239109933628497887838, −4.68022347047030210775763927277, −2.84568155050695396263667364659, −0.74619656443806649656025081230,
1.07493251879020677683581474790, 2.50426655578701489402711116836, 3.35890545463416287192350696956, 4.56448678163307908050769031652, 5.75379356447980563459431787497, 6.33937695944879680577117671629, 7.51733468177736425246789449016, 8.660439772635479706345937693383, 9.629445858522948538067208607225, 10.37890692193416875171004834014
