| L(s) = 1 | + (−0.284 − 1.97i)2-s + (−1.24 + 2.72i)3-s + (−3.83 + 1.12i)4-s + (−11.4 + 13.2i)5-s + (5.75 + 1.69i)6-s + (16.3 − 10.4i)7-s + (3.32 + 7.27i)8-s + (−5.89 − 6.80i)9-s + (29.4 + 18.9i)10-s + (9.06 − 63.0i)11-s + (1.70 − 11.8i)12-s + (−51.1 − 32.8i)13-s + (−25.4 − 29.3i)14-s + (−21.8 − 47.7i)15-s + (13.4 − 8.65i)16-s + (−79.9 − 23.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (−1.02 + 1.18i)5-s + (0.391 + 0.115i)6-s + (0.881 − 0.566i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.930 + 0.598i)10-s + (0.248 − 1.72i)11-s + (0.0410 − 0.285i)12-s + (−1.09 − 0.701i)13-s + (−0.484 − 0.559i)14-s + (−0.375 − 0.821i)15-s + (0.210 − 0.135i)16-s + (−1.14 − 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.284746 - 0.589408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.284746 - 0.589408i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.284 + 1.97i)T \) |
| 3 | \( 1 + (1.24 - 2.72i)T \) |
| 23 | \( 1 + (-24.3 - 107. i)T \) |
| good | 5 | \( 1 + (11.4 - 13.2i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-16.3 + 10.4i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (-9.06 + 63.0i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (51.1 + 32.8i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (79.9 + 23.4i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-114. + 33.7i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (19.4 + 5.72i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (106. + 233. i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (55.5 + 64.0i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-36.2 + 41.8i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-157. + 345. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 + 619.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-108. + 69.7i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (253. + 163. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-125. - 275. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-48.8 - 339. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-63.1 - 439. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-707. + 207. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (1.02e3 + 655. i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-283. - 327. i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (376. - 823. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (636. - 734. i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−11.71114673734284214945884264120, −11.27128230694206646612497273411, −10.73970086792680392590636054238, −9.455104010122351647387530784755, −8.071728658086210132159950606396, −7.17986049034209353358174637122, −5.35402870718496890332159389847, −3.93416450164541979389273091598, −2.96090410420568921015325031234, −0.35838787878227311409214016612,
1.66337371368569300396809873173, 4.58725939574183028639503971513, 4.97526499614574458556286562167, 6.86443725512283566093878827185, 7.71156510448730915115085811274, 8.633051759425539323668349262266, 9.610317688328313371287023805166, 11.42410299390750623739775353992, 12.27215084879301579454788164076, 12.73284926436389528877720409073
