L(s) = 1 | + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + (5.04 + 5.04i)5-s + (−1.34 − 5.02i)7-s + (−2 + 1.99i)8-s + (8.74 − 5.04i)10-s + (7.32 + 1.96i)11-s + (12.9 − 1.42i)13-s − 7.36·14-s + (1.99 + 3.46i)16-s + (13.9 + 8.08i)17-s + (9.87 − 2.64i)19-s + (−3.69 − 13.7i)20-s + (5.36 − 9.28i)22-s + (8.29 − 4.78i)23-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.433 − 0.250i)4-s + (1.00 + 1.00i)5-s + (−0.192 − 0.718i)7-s + (−0.250 + 0.249i)8-s + (0.874 − 0.504i)10-s + (0.665 + 0.178i)11-s + (0.993 − 0.109i)13-s − 0.525·14-s + (0.124 + 0.216i)16-s + (0.823 + 0.475i)17-s + (0.519 − 0.139i)19-s + (−0.184 − 0.689i)20-s + (0.243 − 0.422i)22-s + (0.360 − 0.208i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.91309 - 0.630477i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91309 - 0.630477i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-12.9 + 1.42i)T \) |
good | 5 | \( 1 + (-5.04 - 5.04i)T + 25iT^{2} \) |
| 7 | \( 1 + (1.34 + 5.02i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-7.32 - 1.96i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-13.9 - 8.08i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-9.87 + 2.64i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-8.29 + 4.78i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (16.5 + 28.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (34.2 + 34.2i)T + 961iT^{2} \) |
| 37 | \( 1 + (-63.2 - 16.9i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (14.0 - 52.6i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (40.7 + 23.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (47.8 - 47.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 67.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.4 - 72.7i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (35.2 - 61.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 41.7i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-106. + 28.6i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (36.8 - 36.8i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 13.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (93.1 + 93.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (48.6 + 13.0i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (52.5 - 14.0i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−11.59591413976478076058630843667, −10.90632843583753425202963737042, −9.982995489660396825792324305760, −9.420134679035041116393280831481, −7.86137437130883245837224307612, −6.55694607013509397549126006833, −5.78104556124072354690322328244, −4.07359467749327303983023183852, −2.98002214966656248199241847507, −1.43104049532915552620874427416,
1.43027874335181753694602157070, 3.47971197579200124111141662348, 5.15024145694666370882662666770, 5.71253803402394788286519035165, 6.80121115548165874541389957415, 8.265202615897118741101968781991, 9.137978509544044955403462673813, 9.615240370762823262709340527705, 11.20088633251522130127208347100, 12.38348837061962095462028042408