| L(s) = 1 | − i·2-s − 1.41i·3-s − 4-s − 1.41·6-s − i·7-s + i·8-s + 0.999·9-s + 3.41·11-s + 1.41i·12-s + 3.82i·13-s − 14-s + 16-s + 1.41i·17-s − 0.999i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.816i·3-s − 0.5·4-s − 0.577·6-s − 0.377i·7-s + 0.353i·8-s + 0.333·9-s + 1.02·11-s + 0.408i·12-s + 1.06i·13-s − 0.267·14-s + 0.250·16-s + 0.342i·17-s − 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.912762613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912762613\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 3.82iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 9.07iT - 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 7.24iT - 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 + 0.757iT - 73T^{2} \) |
| 79 | \( 1 - 5.58T + 79T^{2} \) |
| 83 | \( 1 + 7.65iT - 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 - 17.5iT - 97T^{2} \) |
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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
−9.151035049300687337863742049600, −8.137387074885979420337293231953, −7.38319263224584153687254766378, −6.66795242051279451006571298127, −5.95104896992916492072495662003, −4.60433949139618820413667525746, −4.03774999840706421564401706946, −2.97111457244028617562885861556, −1.66696069816880308116917563446, −1.16458072465736565648309825773,
0.848641541044952427111408936370, 2.59514604825645346988188738873, 3.70883091445091825108715958167, 4.47713048544575798849074410149, 5.17202779541679612564311019916, 6.09435630319836451380977712301, 6.80839148050914709950457487411, 7.64511512061600888219732668905, 8.677453472006700538901151231129, 8.962878688753201050428249026675
