| L(s) = 1 | + (−0.494 + 1.32i)2-s + (−1.66 − 0.472i)3-s + (−1.51 − 1.31i)4-s + (−1.70 − 0.621i)5-s + (1.45 − 1.97i)6-s + (3.32 + 0.585i)7-s + (2.48 − 1.35i)8-s + (2.55 + 1.57i)9-s + (1.66 − 1.95i)10-s + (−0.942 − 2.58i)11-s + (1.89 + 2.89i)12-s + (3.98 + 4.74i)13-s + (−2.41 + 4.10i)14-s + (2.55 + 1.84i)15-s + (0.561 + 3.96i)16-s + (2.90 − 1.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.349 + 0.936i)2-s + (−0.962 − 0.272i)3-s + (−0.755 − 0.655i)4-s + (−0.763 − 0.277i)5-s + (0.592 − 0.805i)6-s + (1.25 + 0.221i)7-s + (0.878 − 0.477i)8-s + (0.851 + 0.524i)9-s + (0.527 − 0.617i)10-s + (−0.284 − 0.780i)11-s + (0.547 + 0.836i)12-s + (1.10 + 1.31i)13-s + (−0.646 + 1.09i)14-s + (0.658 + 0.475i)15-s + (0.140 + 0.990i)16-s + (0.703 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.718508 + 0.177528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718508 + 0.177528i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.494 - 1.32i)T \) |
| 3 | \( 1 + (1.66 + 0.472i)T \) |
| good | 5 | \( 1 + (1.70 + 0.621i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.32 - 0.585i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.942 + 2.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 4.74i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.90 + 1.67i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.87 + 4.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.396 + 2.25i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.06 - 5.92i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.38 - 0.244i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.33 + 2.50i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.54 - 1.84i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.42 + 1.24i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.530 + 3.00i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.184T + 53T^{2} \) |
| 59 | \( 1 + (4.44 - 12.2i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (13.9 + 2.45i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.70 + 3.94i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.83 - 3.18i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.136 + 0.236i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.710 + 0.846i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.40 + 2.86i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.59 - 3.22i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.5 - 5.29i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.18182001113077546421169106972, −11.34654175513795731714398248068, −10.71477715018569691230226214857, −9.111143864710181209714386461011, −8.238591479118808401101029339902, −7.37630253948479190556422835237, −6.27398564662480128570050667909, −5.15499813660234681882343297306, −4.33392104650885004925272049379, −1.09426857140388812607180529921,
1.26386817256927369984290598169, 3.53992218023738274858089283403, 4.56534618171351740361363480843, 5.72918240345816971540982739760, 7.74374596998922248518762289365, 7.997600411703911089626015550332, 9.769134979359013066255858331804, 10.56091027715134211697296658567, 11.21728825677287294972541804059, 11.97821520769597337375404731185
