L(s) = 1 | + 2i·3-s + 3i·7-s − 9-s + 5·11-s − 5i·13-s + 4i·17-s − 19-s − 6·21-s − i·23-s + 4i·27-s − 9·29-s − 2·31-s + 10i·33-s + 2i·37-s + 10·39-s + ⋯ |
L(s) = 1 | + 1.15i·3-s + 1.13i·7-s − 0.333·9-s + 1.50·11-s − 1.38i·13-s + 0.970i·17-s − 0.229·19-s − 1.30·21-s − 0.208i·23-s + 0.769i·27-s − 1.67·29-s − 0.359·31-s + 1.74i·33-s + 0.328i·37-s + 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.778738296\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.778738296\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 7iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.909066322925044918388895541115, −8.080804763659231577310421259525, −7.22330535884535179799266917387, −6.09450216737845948746616268209, −5.76136297393160432215938394070, −4.91478241195045360768129282662, −3.99802892059942315797710506709, −3.53633131033904080334257543496, −2.49694239142338475435054588618, −1.33932896517752793247981020920,
0.50674597202377969495523933350, 1.51594355725472460931701144108, 2.09013074114321655717971329401, 3.60907052529555547302216594091, 4.08000153979078475101532858048, 4.99545581530395674895248766149, 6.24038310374922963070809859646, 6.67033661628611123229130692787, 7.27927731261662157156767611314, 7.63822396663609453925233187010