L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 3i·7-s − i·8-s − 9-s − 1.27·11-s + i·12-s − 3.27i·13-s − 3·14-s + 16-s − 2.27i·17-s − i·18-s − 3.27·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.13i·7-s − 0.353i·8-s − 0.333·9-s − 0.384·11-s + 0.288i·12-s − 0.908i·13-s − 0.801·14-s + 0.250·16-s − 0.551i·17-s − 0.235i·18-s − 0.751·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.180871086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180871086\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 + 3.27iT - 13T^{2} \) |
| 17 | \( 1 + 2.27iT - 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 29 | \( 1 - 9.54T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 + 2.72T + 41T^{2} \) |
| 43 | \( 1 + 7.27iT - 43T^{2} \) |
| 47 | \( 1 - 8.27iT - 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 0.549T + 61T^{2} \) |
| 67 | \( 1 + 4.54iT - 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 2.45iT - 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.465849024686079272715110420828, −8.221409177258851023081753942011, −7.37735380184961890147066826696, −6.46337790191808800033049415311, −6.02153536498621179639485852938, −5.17144636821831289427289938401, −4.55032043518056130799547198520, −3.06627010749682571024401609711, −2.53032087339082973469241448771, −1.07410897599337993229649042030,
0.40468141871978687200202296072, 1.71062271864153786691536149519, 2.77454304463137164584763367649, 3.75214447211088061700268482957, 4.37744605085685063959787984480, 4.91046855665867977372712755976, 6.13333043701568385410333876521, 6.79886654172621083134234808096, 7.79557791135833487919264537474, 8.496682801831264748782044265252