L(s) = 1 | + (−0.546 − 1.64i)3-s + (1.08 − 2.41i)7-s + (−2.40 + 1.79i)9-s + (−1.17 − 0.675i)11-s − 4.94i·13-s + (2.87 − 4.97i)17-s + (2.84 − 1.64i)19-s + (−4.55 − 0.460i)21-s + (−4.33 + 2.50i)23-s + (4.26 + 2.96i)27-s + 5.68i·29-s + (−2.45 − 1.41i)31-s + (−0.471 + 2.29i)33-s + (1.92 + 3.33i)37-s + (−8.12 + 2.69i)39-s + ⋯ |
L(s) = 1 | + (−0.315 − 0.948i)3-s + (0.409 − 0.912i)7-s + (−0.801 + 0.598i)9-s + (−0.353 − 0.203i)11-s − 1.37i·13-s + (0.696 − 1.20i)17-s + (0.653 − 0.377i)19-s + (−0.994 − 0.100i)21-s + (−0.903 + 0.521i)23-s + (0.820 + 0.571i)27-s + 1.05i·29-s + (−0.440 − 0.254i)31-s + (−0.0820 + 0.399i)33-s + (0.316 + 0.548i)37-s + (−1.30 + 0.432i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.096927386\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.096927386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.546 + 1.64i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.08 + 2.41i)T \) |
good | 11 | \( 1 + (1.17 + 0.675i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.94iT - 13T^{2} \) |
| 17 | \( 1 + (-2.87 + 4.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.84 + 1.64i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.33 - 2.50i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.68iT - 29T^{2} \) |
| 31 | \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.92 - 3.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 4.06T + 43T^{2} \) |
| 47 | \( 1 + (2.84 + 4.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 0.730i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.34 + 7.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.65 + 0.954i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.51 + 4.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.38iT - 71T^{2} \) |
| 73 | \( 1 + (14.1 + 8.16i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.41 + 4.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + (-8.08 - 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.0iT - 97T^{2} \) |
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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
−8.413551260222198504929628399120, −7.70445435451149274405023041606, −7.40028926873353387589457500194, −6.46521024683273345494290198451, −5.38431788323282882967341134323, −5.06392656392535819159612812501, −3.55794289067514503543957157538, −2.73496289531707022954666443509, −1.37076565854177648256726476934, −0.41706881291630047972420163418,
1.69907149201159378828890180210, 2.79730414899528537109098813651, 4.00542774870664086160288138781, 4.51825803873104768846407742592, 5.75040030690243851661706610816, 5.88113377257010233439111354193, 7.13093191916392343338776107076, 8.244407423807733186891771599110, 8.678465783856670209366047112352, 9.686014173757127419987104499064