| L(s) = 1 | − i·2-s − 1.08i·3-s − 4-s + 1.08i·5-s − 1.08·6-s + i·8-s + 1.82·9-s + 1.08·10-s + (−1.41 + 3i)11-s + 1.08i·12-s − 2.29·13-s + 1.17·15-s + 16-s − 1.82i·18-s + 5.54·19-s − 1.08i·20-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.624i·3-s − 0.5·4-s + 0.484i·5-s − 0.441·6-s + 0.353i·8-s + 0.609·9-s + 0.342·10-s + (−0.426 + 0.904i)11-s + 0.312i·12-s − 0.636·13-s + 0.302·15-s + 0.250·16-s − 0.430i·18-s + 1.27·19-s − 0.242i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596428947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596428947\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 - 3i)T \) |
| good | 3 | \( 1 + 1.08iT - 3T^{2} \) |
| 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 1.75iT - 29T^{2} \) |
| 31 | \( 1 + 1.39iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 4.59T + 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 4.90iT - 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 + 7.20iT - 59T^{2} \) |
| 61 | \( 1 - 5.54T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.56iT - 89T^{2} \) |
| 97 | \( 1 + 8.60iT - 97T^{2} \) |
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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
−9.847592365974424659423344013958, −9.223654019432393341651172800708, −7.945450349312257589760729348968, −7.28998876630746606133754853403, −6.65480953079479363384042798824, −5.25538451243720037337663654789, −4.52277928587640575857705516244, −3.19372051597389921361269959937, −2.31360692479670677200779111040, −1.10187419935215458754528674362,
0.953394461067473168056892469536, 2.93572233355821301836266719111, 4.00505708537280522405248525662, 5.06908838431247316688517118476, 5.40579621433393058305337981063, 6.75907768317048968967174589058, 7.44560416577871807806653590367, 8.408164642381503828420175106391, 9.133552200174162899042104943150, 9.827248705079168064070576392862
