| L(s) = 1 | + (0.732 + 2.73i)2-s + (−12.4 + 7.17i)3-s + (−6.92 + 4i)4-s + (11.3 − 42.5i)5-s + (−28.6 − 28.6i)6-s + (6.42 + 11.1i)7-s + (−16 − 15.9i)8-s + (62.3 − 107. i)9-s + 124.·10-s + 116. i·11-s + (57.3 − 99.3i)12-s + (28.9 − 107. i)13-s + (−25.6 + 25.6i)14-s + (163. + 609. i)15-s + (31.9 − 55.4i)16-s + (501. − 134. i)17-s + ⋯ |
| L(s) = 1 | + (0.183 + 0.683i)2-s + (−1.37 + 0.796i)3-s + (−0.433 + 0.250i)4-s + (0.455 − 1.70i)5-s + (−0.796 − 0.796i)6-s + (0.131 + 0.227i)7-s + (−0.250 − 0.249i)8-s + (0.769 − 1.33i)9-s + 1.24·10-s + 0.959i·11-s + (0.398 − 0.689i)12-s + (0.171 − 0.638i)13-s + (−0.131 + 0.131i)14-s + (0.726 + 2.71i)15-s + (0.124 − 0.216i)16-s + (1.73 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.03860 - 0.0874523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03860 - 0.0874523i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.732 - 2.73i)T \) |
| 37 | \( 1 + (530. - 1.26e3i)T \) |
| good | 3 | \( 1 + (12.4 - 7.17i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-11.3 + 42.5i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-6.42 - 11.1i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 116. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-28.9 + 107. i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-501. + 134. i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-76.0 + 283. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (416. + 416. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-629. + 629. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-556. + 556. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-1.37e3 + 792. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (753. + 753. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 58.2T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.89e3 - 3.28e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (3.91e3 - 1.04e3i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (5.01e3 + 1.34e3i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-7.02e3 + 4.05e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (849. + 1.47e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 4.43e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-2.48e3 + 9.25e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (2.49e3 - 4.32e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.36e3 - 5.07e3i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-1.00e4 - 1.00e4i)T + 8.85e7iT^{2} \) |
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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
−13.75068963917952384426304552278, −12.39396375609185538409173916455, −12.03779558444353269260601895899, −10.19047857837138809891639162770, −9.411862629398332642265396710965, −7.990180581692713192035932150304, −6.07301827283940018493229171506, −5.17093262210718746197997906745, −4.50769659069224807094837622973, −0.68430936731861572091685080190,
1.43367609298006553097633633923, 3.35385228434221662229840256404, 5.66434067494815007027634444193, 6.38914421313315090391060067698, 7.68157804877323493574870988240, 10.00857784908074125042648850998, 10.79646288394919567151774723628, 11.54318138775373216710239506096, 12.45500808128618296565575569074, 13.92534347218265120874531311351
