L(s) = 1 | − 2-s + (−1.68 − 0.420i)3-s + 4-s + (−1.08 + 1.95i)5-s + (1.68 + 0.420i)6-s + (0.595 − 2.57i)7-s − 8-s + (2.64 + 1.41i)9-s + (1.08 − 1.95i)10-s − 2.82i·11-s + (−1.68 − 0.420i)12-s + 3.36·13-s + (−0.595 + 2.57i)14-s + (2.64 − 2.82i)15-s + 16-s − 4.75i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.970 − 0.242i)3-s + 0.5·4-s + (−0.485 + 0.874i)5-s + (0.685 + 0.171i)6-s + (0.224 − 0.974i)7-s − 0.353·8-s + (0.881 + 0.471i)9-s + (0.343 − 0.618i)10-s − 0.852i·11-s + (−0.485 − 0.121i)12-s + 0.931·13-s + (−0.159 + 0.688i)14-s + (0.683 − 0.730i)15-s + 0.250·16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.526091 - 0.280253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.526091 - 0.280253i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.68 + 0.420i)T \) |
| 5 | \( 1 + (1.08 - 1.95i)T \) |
| 7 | \( 1 + (-0.595 + 2.57i)T \) |
good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.75iT - 17T^{2} \) |
| 19 | \( 1 + 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 - 0.500iT - 29T^{2} \) |
| 31 | \( 1 - 3.06iT - 31T^{2} \) |
| 37 | \( 1 - 3.32iT - 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 7.82iT - 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2.52iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 9.81iT - 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 - 6.97iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−11.70785197111325238897137406250, −10.95403705842276213026079908409, −10.80908245693900382438445152298, −9.382343440261723708126784240942, −7.992496374551174770801387784704, −7.04732132802948817631223870605, −6.43895797230488381667563236287, −4.83685746274014639830070849504, −3.19099826398502361242395411892, −0.814560005156893192272222838560,
1.50797735151482452778993416098, 3.98818470394051712313603532686, 5.31438607244306327525566597832, 6.22822123708165516814413525698, 7.64987263477730370528498934760, 8.673767549294495068726572627775, 9.510414568825577933244614762903, 10.67046101683868007527861514649, 11.48640096209861711249102507322, 12.42484092115678720597866305895