L(s) = 1 | − i·2-s − 4-s + (−1 + 2i)5-s + i·7-s + i·8-s + (2 + i)10-s − 5·11-s + 14-s + 16-s − i·17-s + (1 − 2i)20-s + 5i·22-s + 4i·23-s + (−3 − 4i)25-s − i·28-s − 3·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.447 + 0.894i)5-s + 0.377i·7-s + 0.353i·8-s + (0.632 + 0.316i)10-s − 1.50·11-s + 0.267·14-s + 0.250·16-s − 0.242i·17-s + (0.223 − 0.447i)20-s + 1.06i·22-s + 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.188i·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6308448836\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6308448836\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 37 | \( 1 + iT \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + 7iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−8.335161604938855825475387642571, −7.68980325657955844802350644398, −7.10501134937830932904269060839, −5.95775912192599789269712483037, −5.32498614529446413303903557158, −4.38245273044233622472990910587, −3.40363180219163034709022624486, −2.76434728669581474791762038211, −1.95636956043618698503986375288, −0.23854590838559594356905553229,
0.901878546130368232964868803612, 2.37996817428331787590801346737, 3.58905394884931860459991501782, 4.44446803762514620469864485516, 5.09529117194524344805585489636, 5.73778400578310807801356618597, 6.70914310679236794270745960890, 7.56528430257447074955363167617, 8.050300285049835727730483842023, 8.612018417221854147724337855600