| L(s) = 1 | + (−0.830 − 1.81i)2-s + (−2.87 + 0.845i)3-s + (−2.61 + 3.02i)4-s + (3.54 + 2.27i)5-s + (3.92 + 4.53i)6-s + (−0.484 − 3.36i)7-s + (7.67 + 2.25i)8-s + (7.57 − 4.86i)9-s + (1.19 − 8.33i)10-s + (1.72 − 3.78i)11-s + (4.98 − 10.9i)12-s + (5.88 − 40.9i)13-s + (−5.72 + 3.68i)14-s + (−12.1 − 3.55i)15-s + (−2.27 − 15.8i)16-s + (−58.2 − 67.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.316 + 0.203i)5-s + (0.267 + 0.308i)6-s + (−0.0261 − 0.181i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0378 − 0.263i)10-s + (0.0473 − 0.103i)11-s + (0.119 − 0.262i)12-s + (0.125 − 0.873i)13-s + (−0.109 + 0.0702i)14-s + (−0.208 − 0.0612i)15-s + (−0.0355 − 0.247i)16-s + (−0.830 − 0.958i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.384266 - 0.753995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384266 - 0.753995i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 + 1.81i)T \) |
| 3 | \( 1 + (2.87 - 0.845i)T \) |
| 23 | \( 1 + (98.8 + 48.9i)T \) |
| good | 5 | \( 1 + (-3.54 - 2.27i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (0.484 + 3.36i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (-1.72 + 3.78i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 40.9i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (58.2 + 67.2i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-33.8 + 39.0i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (82.2 + 94.9i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (-118. - 34.6i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-58.4 + 37.5i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (115. + 74.3i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-225. + 66.2i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 40.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + (44.7 + 311. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (0.507 - 3.52i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-267. - 78.4i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (-108. - 237. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-121. - 265. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (103. - 119. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (46.6 - 324. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (1.08e3 - 700. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-379. + 111. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (518. + 333. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−12.14393889787306444849923154379, −11.27226638645640119310810747002, −10.35532686848500509055331053344, −9.552701217424508238208570004625, −8.261802565303176524046063468601, −6.91853588968246246167487533586, −5.57468756277586007616007324494, −4.18710624471975914134028242204, −2.53633919039160138879889323100, −0.50897262685012947950237923738,
1.65679660951451930826323589557, 4.19447521818929388578458520030, 5.59609861029680266945397489595, 6.46142878372108573193440258621, 7.65078863506047860664976401769, 8.881784629453061595262958771902, 9.828457862934723650205216453869, 10.99464545039225447451265302193, 12.05631268761149653597790907692, 13.16954079541326625280894324478
