L(s) = 1 | + (−2.58 − 0.455i)2-s + (0.597 − 0.711i)3-s + (4.58 + 1.66i)4-s + (−1.86 + 1.56i)6-s + (3.31 − 1.91i)7-s + (−6.53 − 3.77i)8-s + (0.371 + 2.10i)9-s + (1.63 − 2.82i)11-s + (3.92 − 2.26i)12-s + (−0.234 − 0.278i)13-s + (−9.44 + 3.43i)14-s + (7.67 + 6.44i)16-s + (0.673 + 0.118i)17-s − 5.60i·18-s + (3.86 − 2.01i)19-s + ⋯ |
L(s) = 1 | + (−1.82 − 0.321i)2-s + (0.344 − 0.410i)3-s + (2.29 + 0.833i)4-s + (−0.761 + 0.639i)6-s + (1.25 − 0.724i)7-s + (−2.30 − 1.33i)8-s + (0.123 + 0.701i)9-s + (0.492 − 0.853i)11-s + (1.13 − 0.653i)12-s + (−0.0649 − 0.0773i)13-s + (−2.52 + 0.918i)14-s + (1.91 + 1.61i)16-s + (0.163 + 0.0288i)17-s − 1.32i·18-s + (0.886 − 0.461i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.737683 - 0.399138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.737683 - 0.399138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.86 + 2.01i)T \) |
good | 2 | \( 1 + (2.58 + 0.455i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.597 + 0.711i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 1.91i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.234 + 0.278i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.673 - 0.118i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (3.20 - 8.80i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.267 + 1.51i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 2.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.163iT - 37T^{2} \) |
| 41 | \( 1 + (5.64 + 4.73i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.885 + 2.43i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-8.66 + 1.52i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.80 + 10.4i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.35 + 7.67i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.27 + 0.827i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (9.41 - 1.65i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.81 + 1.75i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-5.06 + 6.03i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.14 - 5.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.631 + 0.364i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.68 - 8.12i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.43 - 1.31i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.79014509720200775232520813813, −9.943785580166521678224464274797, −8.961854684911540426125798283399, −8.121542993639056595685130727077, −7.65140893754220185182079061196, −6.90362530610049355559009348345, −5.32912716818812299077474274582, −3.52379843411662773843584690481, −2.02730032144123225159793192703, −1.09715928305547019981481795624,
1.34541370551421760268195673533, 2.55901801757652202144526932988, 4.44117632239520773800923755445, 5.87613813912106350832600283677, 6.89377896142144340620526631203, 7.86053088140741215846730621833, 8.578788487497417659681368969714, 9.246980206485855113942407196946, 9.983707441542191406168175082166, 10.75521935665056912894536261533