L(s) = 1 | + 3i·3-s − 20.7·5-s + (11.8 − 14.2i)7-s − 9·9-s − 39.2·11-s + 36.2·13-s − 62.1i·15-s + 92.2i·17-s + 87.5i·19-s + (42.7 + 35.4i)21-s + 100. i·23-s + 303.·25-s − 27i·27-s − 36.2i·29-s + 264.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.85·5-s + (0.638 − 0.770i)7-s − 0.333·9-s − 1.07·11-s + 0.773·13-s − 1.06i·15-s + 1.31i·17-s + 1.05i·19-s + (0.444 + 0.368i)21-s + 0.915i·23-s + 2.42·25-s − 0.192i·27-s − 0.231i·29-s + 1.53·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1670904010\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1670904010\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (-11.8 + 14.2i)T \) |
good | 5 | \( 1 + 20.7T + 125T^{2} \) |
| 11 | \( 1 + 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 87.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 100. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 36.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 229. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 337. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 108. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 185. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 584. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 438. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 722. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 452. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 597. iT - 9.12e5T^{2} \) |
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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
−8.573140551813244472568165138050, −8.142818161152921780513585135899, −7.71276536269013796941703676432, −6.63871404298660382934732537964, −5.42330804928549163690187396044, −4.46876305075423389911106937869, −3.86734037062241388842175981337, −3.18967278825928807864052668006, −1.38332870538310031943018758759, −0.05252625903644881497038222007,
0.874119594320551316532076012751, 2.53262510177459466259288501169, 3.21913864501775287464267843720, 4.61870933591328217304161020801, 5.03142724505610399072843621913, 6.40630663594204314936552518210, 7.21969244160346972865928401435, 8.048415338099462068410598022265, 8.326585491410504035680693978721, 9.190886481703701696244682546849