| L(s) = 1 | + (0.599 + 1.44i)2-s + (1.44 − 0.962i)3-s + (1.09 − 1.09i)4-s + (1.92 + 0.383i)5-s + (2.25 + 1.50i)6-s + (−10.3 + 2.06i)7-s + (8.02 + 3.32i)8-s + (1.14 − 2.77i)9-s + (0.600 + 3.01i)10-s + (−6.35 + 9.50i)11-s + (0.523 − 2.63i)12-s + (−9.01 − 9.01i)13-s + (−9.18 − 13.7i)14-s + (3.14 − 1.30i)15-s + 7.40i·16-s + (5.71 − 16.0i)17-s + ⋯ |
| L(s) = 1 | + (0.299 + 0.723i)2-s + (0.480 − 0.320i)3-s + (0.273 − 0.273i)4-s + (0.385 + 0.0767i)5-s + (0.375 + 0.251i)6-s + (−1.47 + 0.294i)7-s + (1.00 + 0.415i)8-s + (0.127 − 0.307i)9-s + (0.0600 + 0.301i)10-s + (−0.577 + 0.864i)11-s + (0.0436 − 0.219i)12-s + (−0.693 − 0.693i)13-s + (−0.656 − 0.981i)14-s + (0.209 − 0.0868i)15-s + 0.462i·16-s + (0.336 − 0.941i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46977 + 0.364923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46977 + 0.364923i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.44 + 0.962i)T \) |
| 17 | \( 1 + (-5.71 + 16.0i)T \) |
| good | 2 | \( 1 + (-0.599 - 1.44i)T + (-2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (-1.92 - 0.383i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (10.3 - 2.06i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (6.35 - 9.50i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (9.01 + 9.01i)T + 169iT^{2} \) |
| 19 | \( 1 + (1.81 + 4.37i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-9.60 - 6.41i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (6.45 - 32.4i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (-27.5 - 41.2i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-43.9 + 29.3i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (28.1 - 5.60i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (-13.1 + 31.6i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-16.3 - 16.3i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-11.3 - 27.4i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (5.30 + 2.19i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (19.8 + 99.8i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 6.18iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (44.9 - 30.0i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-60.9 - 12.1i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (50.9 - 76.1i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-47.8 + 19.8i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-104. + 104. i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-10.9 + 54.9i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−15.43539617068565374795929433896, −14.32906293666169920474136797467, −13.25962151377665455315042129392, −12.32534506162235825896343320717, −10.35156995537747792975580954320, −9.448235065383364267094794906868, −7.56218101485967586261066535297, −6.63147751002009792783874158354, −5.26715653357291585271571582281, −2.70008117990903072569770570445,
2.66782274489159443667314416739, 3.98025417527282799428555145161, 6.25830711467882415958109361764, 7.85658448218388039978530297489, 9.586466836918695916857468290494, 10.37756861012663935374262226375, 11.78443628827426865497391673127, 13.11801245152634707646939359070, 13.53647873196525220292756704892, 15.19529385396951610307856995719
