L(s) = 1 | + (0.129 − 0.0745i)2-s + (3.24 + 5.61i)3-s + (−3.98 + 6.90i)4-s − 10.2i·5-s + (0.837 + 0.483i)6-s + (−25.6 − 14.8i)7-s + 2.38i·8-s + (−7.55 + 13.0i)9-s + (−0.764 − 1.32i)10-s + (−32.9 + 19.0i)11-s − 51.7·12-s − 4.42·14-s + (57.6 − 33.2i)15-s + (−31.7 − 54.9i)16-s + (−35.6 + 61.7i)17-s + 2.25i·18-s + ⋯ |
L(s) = 1 | + (0.0456 − 0.0263i)2-s + (0.624 + 1.08i)3-s + (−0.498 + 0.863i)4-s − 0.917i·5-s + (0.0569 + 0.0329i)6-s + (−1.38 − 0.801i)7-s + 0.105i·8-s + (−0.279 + 0.484i)9-s + (−0.0241 − 0.0418i)10-s + (−0.904 + 0.522i)11-s − 1.24·12-s − 0.0844·14-s + (0.991 − 0.572i)15-s + (−0.495 − 0.858i)16-s + (−0.508 + 0.880i)17-s + 0.0294i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0394996 - 0.119447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0394996 - 0.119447i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-0.129 + 0.0745i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.24 - 5.61i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + 10.2iT - 125T^{2} \) |
| 7 | \( 1 + (25.6 + 14.8i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (32.9 - 19.0i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (35.6 - 61.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8.74 + 5.04i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (99.3 + 172. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (15.4 + 26.6i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 151. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (131. - 75.6i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (179. - 103. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (151. - 262. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 12.2iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-338. - 195. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-78.2 + 135. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (263. - 151. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (791. + 456. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 - 249. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-819. + 473. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (361. + 208. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.88235232443863894527394030312, −12.37973685421506420848011142604, −10.47029862344170271398827827243, −9.867539948207911932418137730909, −8.869546657155417516321815756509, −8.173793543970602710288374316630, −6.70202012119615212957538999719, −4.77047707732924287736455333331, −4.05883700330168856838472399786, −2.98180667248941052317198166115,
0.04980980498840842621380763588, 2.18471356752489482811119463321, 3.27129695174990985054150670272, 5.49195926095208368339458665862, 6.43655463550733485133824773056, 7.35896304336122111223001609233, 8.684807011431600042428256702247, 9.642332852999105282443149186475, 10.56612986706277597226919040094, 11.87498869572690177627234260347