L(s) = 1 | + (1.28 − 0.599i)2-s + (1.66 + 0.468i)3-s + (1.28 − 1.53i)4-s + (2.41 − 0.400i)6-s − 0.936·7-s + (0.719 − 2.73i)8-s + (2.56 + 1.56i)9-s − 4.27·11-s + (2.85 − 1.96i)12-s + 3.12i·13-s + (−1.19 + 0.561i)14-s + (−0.719 − 3.93i)16-s − 2·17-s + (4.21 + 0.463i)18-s − 4.27i·19-s + ⋯ |
L(s) = 1 | + (0.905 − 0.424i)2-s + (0.962 + 0.270i)3-s + (0.640 − 0.768i)4-s + (0.986 − 0.163i)6-s − 0.353·7-s + (0.254 − 0.967i)8-s + (0.853 + 0.520i)9-s − 1.28·11-s + (0.824 − 0.566i)12-s + 0.866i·13-s + (−0.320 + 0.150i)14-s + (−0.179 − 0.983i)16-s − 0.485·17-s + (0.994 + 0.109i)18-s − 0.979i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.56232 - 0.661211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.56232 - 0.661211i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.28 + 0.599i)T \) |
| 3 | \( 1 + (-1.66 - 0.468i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.936T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 3.12iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4.27iT - 19T^{2} \) |
| 23 | \( 1 - 7.60iT - 23T^{2} \) |
| 29 | \( 1 + 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 3.12iT - 37T^{2} \) |
| 41 | \( 1 + 7.12iT - 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 - 0.936iT - 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 5.20T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 + 8.24iT - 73T^{2} \) |
| 79 | \( 1 - 9.06iT - 79T^{2} \) |
| 83 | \( 1 + 4.68iT - 83T^{2} \) |
| 89 | \( 1 + 6.24iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−11.70635896202546437086121982292, −10.77426482509852337946012285360, −9.867897376675035580056677581525, −9.065087473140376063948904252416, −7.70235677392549845145314095871, −6.77543910182578644308898838301, −5.33861784205783069771007118629, −4.32097337955188654482571814383, −3.15858736582851018107682320876, −2.10971197066226745619528438226,
2.39849004701654073599190469424, 3.33195707669791873931256695076, 4.62041650311635729780935191155, 5.87534350301891697550193613195, 6.97787043199817082966856775787, 7.941649147637576721726949880687, 8.543309544780603672278905884271, 10.01847134508176820449332796813, 10.88496300443369316536086823613, 12.40899137443446028492979818768