| L(s) = 1 | + (−1.52 − 0.285i)2-s + (0.0891 + 0.179i)3-s + (0.380 + 0.147i)4-s + (2.68 − 1.66i)5-s + (−0.0849 − 0.298i)6-s + (−3.14 + 4.16i)7-s + (2.10 + 1.30i)8-s + (1.78 − 2.36i)9-s + (−4.56 + 1.76i)10-s + (1.90 + 0.736i)11-s + (0.00752 + 0.0811i)12-s + (−0.700 − 0.433i)13-s + (5.98 − 5.45i)14-s + (0.536 + 0.332i)15-s + (−3.43 − 3.13i)16-s + (−1.83 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−1.07 − 0.201i)2-s + (0.0514 + 0.103i)3-s + (0.190 + 0.0736i)4-s + (1.19 − 0.742i)5-s + (−0.0346 − 0.121i)6-s + (−1.18 + 1.57i)7-s + (0.742 + 0.459i)8-s + (0.594 − 0.787i)9-s + (−1.44 + 0.559i)10-s + (0.572 + 0.221i)11-s + (0.00217 + 0.0234i)12-s + (−0.194 − 0.120i)13-s + (1.59 − 1.45i)14-s + (0.138 + 0.0857i)15-s + (−0.858 − 0.783i)16-s + (−0.445 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.845684 - 0.0408847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.845684 - 0.0408847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (1.83 - 3.69i)T \) |
| good | 2 | \( 1 + (1.52 + 0.285i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (-0.0891 - 0.179i)T + (-1.80 + 2.39i)T^{2} \) |
| 5 | \( 1 + (-2.68 + 1.66i)T + (2.22 - 4.47i)T^{2} \) |
| 7 | \( 1 + (3.14 - 4.16i)T + (-1.91 - 6.73i)T^{2} \) |
| 11 | \( 1 + (-1.90 - 0.736i)T + (8.12 + 7.41i)T^{2} \) |
| 13 | \( 1 + (0.700 + 0.433i)T + (5.79 + 11.6i)T^{2} \) |
| 19 | \( 1 + (-7.03 + 1.31i)T + (17.7 - 6.86i)T^{2} \) |
| 23 | \( 1 + (-3.19 + 4.23i)T + (-6.29 - 22.1i)T^{2} \) |
| 29 | \( 1 + (-4.91 - 1.90i)T + (21.4 + 19.5i)T^{2} \) |
| 31 | \( 1 + (-8.86 - 5.48i)T + (13.8 + 27.7i)T^{2} \) |
| 37 | \( 1 + (-0.579 + 6.25i)T + (-36.3 - 6.79i)T^{2} \) |
| 41 | \( 1 + (1.04 + 2.09i)T + (-24.7 + 32.7i)T^{2} \) |
| 43 | \( 1 + (2.45 - 2.23i)T + (3.96 - 42.8i)T^{2} \) |
| 47 | \( 1 + (1.31 + 1.73i)T + (-12.8 + 45.2i)T^{2} \) |
| 53 | \( 1 + (5.36 - 7.10i)T + (-14.5 - 50.9i)T^{2} \) |
| 59 | \( 1 + (0.471 - 1.65i)T + (-50.1 - 31.0i)T^{2} \) |
| 61 | \( 1 + (-0.287 - 1.00i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (8.02 - 1.49i)T + (62.4 - 24.2i)T^{2} \) |
| 71 | \( 1 + (5.43 - 7.19i)T + (-19.4 - 68.2i)T^{2} \) |
| 73 | \( 1 + (3.10 + 2.82i)T + (6.73 + 72.6i)T^{2} \) |
| 79 | \( 1 + (8.17 - 1.52i)T + (73.6 - 28.5i)T^{2} \) |
| 83 | \( 1 + (-0.692 + 1.39i)T + (-50.0 - 66.2i)T^{2} \) |
| 89 | \( 1 + (5.54 - 3.43i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (-6.86 + 9.09i)T + (-26.5 - 93.2i)T^{2} \) |
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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
−11.91908023211623733893813946769, −10.25945977859771351495960365846, −9.703790469229761384946001494593, −9.032421372402371770505404350544, −8.674955028705798191682717188513, −6.83483304973578300023785428261, −5.91950164789998476738457316205, −4.80575015041953903508180250548, −2.79992445230580208852199691003, −1.31473287334011632210106657014,
1.19010935565912001460671595280, 3.10747948035854071433735285548, 4.61255309006409581086463592763, 6.41892943071717963461614217798, 7.04612119093309298943578600578, 7.78438225429168550181218057308, 9.405519460276273133035022470305, 9.921649204000562441473251317474, 10.26899792892774265169067258025, 11.44906906610834735847593857545
