| L(s) = 1 | + (−3.61 + 1.70i)2-s + (10.1 − 12.3i)4-s + (2.83 − 4.90i)5-s + (−45.1 + 26.0i)7-s + (−15.6 + 62.0i)8-s + (−1.85 + 22.5i)10-s + (92.3 − 53.3i)11-s + (61.0 − 105. i)13-s + (118. − 171. i)14-s + (−49.5 − 251. i)16-s + 122.·17-s + 593. i·19-s + (−31.8 − 84.8i)20-s + (−242. + 350. i)22-s + (473. + 273. i)23-s + ⋯ |
| L(s) = 1 | + (−0.904 + 0.427i)2-s + (0.634 − 0.772i)4-s + (0.113 − 0.196i)5-s + (−0.921 + 0.532i)7-s + (−0.244 + 0.969i)8-s + (−0.0185 + 0.225i)10-s + (0.763 − 0.440i)11-s + (0.361 − 0.625i)13-s + (0.605 − 0.874i)14-s + (−0.193 − 0.981i)16-s + 0.424·17-s + 1.64i·19-s + (−0.0796 − 0.212i)20-s + (−0.501 + 0.724i)22-s + (0.895 + 0.516i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.906533 + 0.517486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.906533 + 0.517486i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.61 - 1.70i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.83 + 4.90i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (45.1 - 26.0i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-92.3 + 53.3i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-61.0 + 105. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 122.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 593. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-473. - 273. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-367. - 637. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (507. + 292. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 2.28e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (1.43e3 - 2.48e3i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.94e3 + 1.12e3i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (913. - 527. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 4.75e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.86e3 + 1.07e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-33.1 - 57.4i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.55e3 + 2.05e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 5.03e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.70e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.19e3 + 690. i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-2.60e3 + 1.50e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 3.18e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-2.40e3 - 4.16e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−13.10807529804070251876393810702, −12.00535049252811668833082006969, −10.82186166333891600505146089241, −9.688232687768417232950213390196, −8.935775866740767096680276248900, −7.77655575682107198644305689316, −6.39739025999097521274756494357, −5.55912426324291342451980842717, −3.22181921741162671277183338828, −1.19045631746223363094626627644,
0.76166590474402422530030790468, 2.66960433231473273589000474454, 4.13713111554837133701954710466, 6.51531096675121698974037482262, 7.15564462096576989159206703329, 8.777047873406504894935781193040, 9.576002331467153599881974572261, 10.57539937180594399341064195573, 11.57618357974080711384631503399, 12.65989399011706538757169809083
