| L(s) = 1 | + (−1.29 − 0.566i)2-s + (0.502 + 3.33i)3-s + (1.35 + 1.46i)4-s + (−0.588 − 0.231i)5-s + (1.23 − 4.60i)6-s + (−1.76 − 1.97i)7-s + (−0.930 − 2.67i)8-s + (−7.98 + 2.46i)9-s + (0.632 + 0.632i)10-s + (−0.526 + 1.70i)11-s + (−4.20 + 5.26i)12-s + (−0.264 + 0.0603i)13-s + (1.16 + 3.55i)14-s + (0.474 − 2.07i)15-s + (−0.305 + 3.98i)16-s + (−4.13 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.916 − 0.400i)2-s + (0.290 + 1.92i)3-s + (0.679 + 0.733i)4-s + (−0.263 − 0.103i)5-s + (0.504 − 1.87i)6-s + (−0.666 − 0.745i)7-s + (−0.329 − 0.944i)8-s + (−2.66 + 0.821i)9-s + (0.199 + 0.200i)10-s + (−0.158 + 0.514i)11-s + (−1.21 + 1.52i)12-s + (−0.0733 + 0.0167i)13-s + (0.311 + 0.950i)14-s + (0.122 − 0.536i)15-s + (−0.0764 + 0.997i)16-s + (−1.00 + 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0461745 - 0.345847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0461745 - 0.345847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 + 0.566i)T \) |
| 7 | \( 1 + (1.76 + 1.97i)T \) |
| good | 3 | \( 1 + (-0.502 - 3.33i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (0.588 + 0.231i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.526 - 1.70i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (0.264 - 0.0603i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (4.13 - 2.82i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.56 + 0.904i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.95 + 1.33i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (4.18 - 8.68i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.51 + 4.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.463 + 0.0346i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-3.40 - 4.27i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.05 - 4.82i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-6.04 + 5.61i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.237 + 0.0177i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (9.81 - 3.85i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (9.69 - 0.726i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-5.49 - 3.16i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.3 - 4.97i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.95 - 1.81i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-6.84 - 11.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.07 + 1.61i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (6.70 - 2.06i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−11.20479681999241389622595781680, −10.68127907102264512588505852794, −9.927955835437419603604574310552, −9.335811382203938708044899363504, −8.515065117181918861174259473917, −7.48406161433040252644918248321, −6.08647778179797768721170388119, −4.45209882631485041217880755320, −3.81650145616187414268785161031, −2.65626315552896429854612310827,
0.27501510631500181687306193799, 1.98547367263638457008585918079, 2.95762672201503577382008446698, 5.73372404900967687830073890318, 6.24195195417757415801148222806, 7.29198346229513530955538014690, 7.82340463029881293972373727033, 8.826374016913125891934131148958, 9.419909983912670634207927763127, 11.03040374104443934691224154054
