| L(s) = 1 | + (−0.675 + 1.63i)2-s + (−3.36 − 2.24i)3-s + (0.624 + 0.624i)4-s + (−7.58 + 1.50i)5-s + (5.94 − 3.96i)6-s + (−4.01 − 0.798i)7-s + (−7.96 + 3.29i)8-s + (2.82 + 6.81i)9-s + (2.66 − 13.3i)10-s + (3.85 + 5.77i)11-s + (−0.697 − 3.50i)12-s + (−2.37 + 2.37i)13-s + (4.01 − 6.00i)14-s + (28.9 + 11.9i)15-s − 11.6i·16-s + ⋯ |
| L(s) = 1 | + (−0.337 + 0.815i)2-s + (−1.12 − 0.749i)3-s + (0.156 + 0.156i)4-s + (−1.51 + 0.301i)5-s + (0.990 − 0.661i)6-s + (−0.573 − 0.114i)7-s + (−0.995 + 0.412i)8-s + (0.313 + 0.757i)9-s + (0.266 − 1.33i)10-s + (0.350 + 0.524i)11-s + (−0.0581 − 0.292i)12-s + (−0.182 + 0.182i)13-s + (0.286 − 0.429i)14-s + (1.92 + 0.798i)15-s − 0.730i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0637i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.435225 - 0.0138770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435225 - 0.0138770i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (0.675 - 1.63i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (3.36 + 2.24i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (7.58 - 1.50i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (4.01 + 0.798i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-3.85 - 5.77i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (2.37 - 2.37i)T - 169iT^{2} \) |
| 19 | \( 1 + (-9.43 + 22.7i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-9.57 + 6.40i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-2.34 - 11.7i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-21.0 + 31.4i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-26.8 - 17.9i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-33.1 - 6.59i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-1.33 - 3.21i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (3.16 - 3.16i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (11.7 - 28.3i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-10.8 + 4.49i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-13.8 + 69.5i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 28.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-28.5 - 19.0i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (91.8 - 18.2i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-11.9 - 17.8i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (104. + 43.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-28.3 - 28.3i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (32.4 + 163. i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.53254033679320882332009528111, −11.16910329009987271160464016372, −9.526963785857378897884063594993, −8.292244974934896962019479887064, −7.24873476004431489477705973916, −6.97604377869381240939717271412, −6.03821970274723652682374689214, −4.55702224446205081764446394165, −3.03030318548621011436026237179, −0.43060117654496251862540242158,
0.791566606905266164325046499722, 3.20647529124500709640903055039, 4.16550727090641607178471530793, 5.48699082390709938090410469308, 6.47077743051346503521482529971, 7.84742386699406226952959120825, 9.067737168552739575738596368174, 10.05291924771225474876442920104, 10.78210469550571128024680397613, 11.63394616832544125599371494511
