L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.766 + 1.32i)5-s + (−0.173 + 0.300i)7-s − 0.999·8-s + 1.53·10-s + (−0.5 − 0.866i)13-s + (0.173 + 0.300i)14-s + (−0.5 + 0.866i)16-s + 1.87·17-s + (0.766 − 1.32i)20-s + (−0.673 + 1.16i)25-s − 0.999·26-s + 0.347·28-s + (0.5 + 0.866i)31-s + (0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.766 + 1.32i)5-s + (−0.173 + 0.300i)7-s − 0.999·8-s + 1.53·10-s + (−0.5 − 0.866i)13-s + (0.173 + 0.300i)14-s + (−0.5 + 0.866i)16-s + 1.87·17-s + (0.766 − 1.32i)20-s + (−0.673 + 1.16i)25-s − 0.999·26-s + 0.347·28-s + (0.5 + 0.866i)31-s + (0.499 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.699459007\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.699459007\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 0.347T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.421853164877690230736000855764, −8.149093408070892003222491639418, −7.34240479716333537416330302585, −6.35107774991621668797622593955, −5.76602233670483120960436197330, −5.16119840390072679833862102174, −3.91216246553446994489744036426, −2.83908497302343051494416634656, −2.75557608694147840339422236317, −1.32895126404561611978888288606,
1.11340207452450151835295177409, 2.52771322891463967329343798997, 3.76700983914554452879312746881, 4.55237577424930014543191498964, 5.24045570336551185413641941502, 5.87828508162788075394050923481, 6.61186611111138613464968857565, 7.63625073741670834978923810334, 8.106899369410880587548614385003, 9.021073997213878036094775919606