L(s) = 1 | + (−1.73 − i)3-s + 4i·7-s + (0.499 + 0.866i)9-s − 3·11-s + (5.19 − 3i)13-s + (1.73 + i)17-s + (−3.5 − 2.59i)19-s + (4 − 6.92i)21-s + (3.46 − 2i)23-s + 4.00i·27-s + (0.5 + 0.866i)29-s − 5·31-s + (5.19 + 3i)33-s + 4i·37-s − 12·39-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.577i)3-s + 1.51i·7-s + (0.166 + 0.288i)9-s − 0.904·11-s + (1.44 − 0.832i)13-s + (0.420 + 0.242i)17-s + (−0.802 − 0.596i)19-s + (0.872 − 1.51i)21-s + (0.722 − 0.417i)23-s + 0.769i·27-s + (0.0928 + 0.160i)29-s − 0.898·31-s + (0.904 + 0.522i)33-s + 0.657i·37-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0379 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0379 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6731027595\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6731027595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.5 + 2.59i)T \) |
good | 3 | \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-5.19 + 3i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.1 + 7i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (10.3 + 6i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + (-8.5 - 14.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 - 6i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.246110864950105460330260309693, −8.562976960403367275194276757733, −7.964713474535941505454066573908, −6.79072139626187181659625854406, −6.10714371964561215871727989540, −5.56379409686943330221828030992, −4.92898894419454928772284863214, −3.39158629347785118188882476289, −2.47654431552578436207129508419, −1.15936733279726361671232939108,
0.31738247707135691938915555307, 1.67857715275242319453313053644, 3.42111802997040223553139115874, 4.12841700320055809496326994545, 4.91096878541930056402387134607, 5.77153856994110900655094108920, 6.53778704800134409740544086563, 7.37743915904621373835084128901, 8.159336276311942147928200648921, 9.107630547627737890228495305881