L(s) = 1 | + (−0.409 + 1.35i)2-s − 3-s + (−1.66 − 1.10i)4-s − 3.34·5-s + (0.409 − 1.35i)6-s + 4.25·7-s + (2.18 − 1.79i)8-s + 9-s + (1.37 − 4.52i)10-s − 1.35i·11-s + (1.66 + 1.10i)12-s + 2.36i·13-s + (−1.74 + 5.75i)14-s + 3.34·15-s + (1.53 + 3.69i)16-s − 1.28i·17-s + ⋯ |
L(s) = 1 | + (−0.289 + 0.957i)2-s − 0.577·3-s + (−0.832 − 0.554i)4-s − 1.49·5-s + (0.167 − 0.552i)6-s + 1.60·7-s + (0.772 − 0.635i)8-s + 0.333·9-s + (0.433 − 1.43i)10-s − 0.409i·11-s + (0.480 + 0.320i)12-s + 0.656i·13-s + (−0.465 + 1.53i)14-s + 0.863·15-s + (0.384 + 0.923i)16-s − 0.312i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.400400 + 0.617054i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.400400 + 0.617054i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.409 - 1.35i)T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + (1.27 - 4.62i)T \) |
good | 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 + 1.35iT - 11T^{2} \) |
| 13 | \( 1 - 2.36iT - 13T^{2} \) |
| 17 | \( 1 + 1.28iT - 17T^{2} \) |
| 19 | \( 1 + 1.70iT - 19T^{2} \) |
| 29 | \( 1 - 7.24iT - 29T^{2} \) |
| 31 | \( 1 - 9.70iT - 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 6.05iT - 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 7.81T + 59T^{2} \) |
| 61 | \( 1 - 6.82T + 61T^{2} \) |
| 67 | \( 1 + 7.98iT - 67T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 - 6.03T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 0.0200iT - 83T^{2} \) |
| 89 | \( 1 - 2.91iT - 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−11.37134686230506915744663656789, −10.26542996199122806580766839487, −8.852433812197868022274973017241, −8.306463363795926884264018001601, −7.43429750657491677240966196567, −6.85202409757424253367490418883, −5.35500580672450336288112687773, −4.74999908915888570011544948690, −3.80172307411665962304848890537, −1.21451008784784854674854590278,
0.62501962619615271566617258735, 2.23391664484069547603017110513, 3.99718517357624639176352174969, 4.41475559799431217199630526416, 5.54624733796898942896647605323, 7.35705366896405217965373511794, 8.018899522425658328719815595070, 8.503563379750239277340294844976, 9.966693520294487692646423437006, 10.76458127428804157301431659168