| L(s) = 1 | + (3.23 + 2.35i)2-s + 22.2i·3-s + (4.94 + 15.2i)4-s + (20.2 + 62.2i)5-s + (−52.3 + 71.9i)6-s + (31.0 + 42.7i)7-s + (−19.7 + 60.8i)8-s − 251.·9-s + (−80.8 + 248. i)10-s + (172. + 56.0i)11-s + (−338. + 109. i)12-s + (502. − 690. i)13-s + 211. i·14-s + (−1.38e3 + 449. i)15-s + (−207. + 150. i)16-s + (347. + 112. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + 1.42i·3-s + (0.154 + 0.475i)4-s + (0.361 + 1.11i)5-s + (−0.593 + 0.816i)6-s + (0.239 + 0.329i)7-s + (−0.109 + 0.336i)8-s − 1.03·9-s + (−0.255 + 0.787i)10-s + (0.429 + 0.139i)11-s + (−0.678 + 0.220i)12-s + (0.823 − 1.13i)13-s + 0.288i·14-s + (−1.58 + 0.516i)15-s + (−0.202 + 0.146i)16-s + (0.291 + 0.0946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.335878 + 2.63212i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335878 + 2.63212i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 41 | \( 1 + (-1.02e4 - 3.39e3i)T \) |
| good | 3 | \( 1 - 22.2iT - 243T^{2} \) |
| 5 | \( 1 + (-20.2 - 62.2i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-31.0 - 42.7i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 11 | \( 1 + (-172. - 56.0i)T + (1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-502. + 690. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-347. - 112. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (762. + 1.04e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (713. + 518. i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.60e3 - 847. i)T + (1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.61e3 + 4.95e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-3.06e3 - 9.42e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 43 | \( 1 + (2.04e3 + 1.48e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (4.67e3 - 6.43e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.30e4 - 4.22e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.11e4 - 8.11e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.83e4 + 2.06e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (2.85e4 - 9.28e3i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-5.50e4 - 1.79e4i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 - 8.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.31e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.60e3 + 2.20e3i)T + (-1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.15e5 - 3.75e4i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.25004799752433052009630040726, −12.99863997962795606468016398630, −11.37699985890894430532312493417, −10.61883865349846066554752882120, −9.585102740201855653482232998597, −8.198527093838204328831028347556, −6.53075402456425186193473072363, −5.41775627163939668430069944937, −4.02665972781014654077668588620, −2.85080460538011753090005587529,
1.02885556308975373577911875207, 1.86376915349329140441149359330, 4.11045554906379320031798630625, 5.70472603145884687024110707036, 6.80390667682001055897232659709, 8.215865191272190837995193757329, 9.368513018793841925525353593487, 11.09695235297606760920728407564, 12.15529834029403219795867313385, 12.81451274160022677636647018629
