L(s) = 1 | − 5-s − 1.41i·7-s − 11-s − 13-s − 17-s − 19-s + 23-s − 1.41i·31-s + 1.41i·35-s + 1.41i·37-s − 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯ |
L(s) = 1 | − 5-s − 1.41i·7-s − 11-s − 13-s − 17-s − 19-s + 23-s − 1.41i·31-s + 1.41i·35-s + 1.41i·37-s − 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3823805483\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3823805483\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.798387357837920282483225843388, −8.571124003580642402683794729419, −7.86793445908189252636694017840, −7.22604201157980910646828305987, −6.55652904619864707357625669315, −5.05062291657153624056412159918, −4.39834475904926607730509203379, −3.55991240279185510432456386374, −2.29341822991056717597436260083, −0.30799226672292309201947228854,
2.24228587948386791577121929546, 2.98246259987145635389225049074, 4.36632991453827353164468195972, 5.09168462835072136516363774183, 6.01753732011591840324226853568, 7.11853972412115120464203667236, 7.80028273994966925569329579545, 8.762271147200267393356025636917, 9.108580759732845590274149584498, 10.39767306844368612725729412196