| L(s) = 1 | + (−1.38 + 0.297i)2-s + (0.428 − 1.67i)3-s + (1.82 − 0.823i)4-s + (0.606 − 1.99i)5-s + (−0.0922 + 2.44i)6-s + (−0.441 − 2.22i)7-s + (−2.27 + 1.68i)8-s + (−2.63 − 1.43i)9-s + (−0.242 + 2.94i)10-s + (4.48 − 0.441i)11-s + (−0.601 − 3.41i)12-s + (−0.494 − 1.63i)13-s + (1.27 + 2.93i)14-s + (−3.09 − 1.87i)15-s + (2.64 − 3.00i)16-s + (−2.01 − 0.835i)17-s + ⋯ |
| L(s) = 1 | + (−0.977 + 0.210i)2-s + (0.247 − 0.968i)3-s + (0.911 − 0.411i)4-s + (0.271 − 0.893i)5-s + (−0.0376 + 0.999i)6-s + (−0.167 − 0.839i)7-s + (−0.804 + 0.594i)8-s + (−0.877 − 0.479i)9-s + (−0.0767 + 0.930i)10-s + (1.35 − 0.133i)11-s + (−0.173 − 0.984i)12-s + (−0.137 − 0.452i)13-s + (0.340 + 0.785i)14-s + (−0.798 − 0.483i)15-s + (0.660 − 0.750i)16-s + (−0.489 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436203 - 0.807611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436203 - 0.807611i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 0.297i)T \) |
| 3 | \( 1 + (-0.428 + 1.67i)T \) |
| good | 5 | \( 1 + (-0.606 + 1.99i)T + (-4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (0.441 + 2.22i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-4.48 + 0.441i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (0.494 + 1.63i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (2.01 + 0.835i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (4.00 - 7.50i)T + (-10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (0.408 + 0.611i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-5.82 - 0.573i)T + (28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-0.361 - 0.361i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.66 + 6.84i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (5.77 + 8.63i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (8.33 + 6.83i)T + (8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-11.8 - 4.90i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.630 + 0.0621i)T + (51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (1.27 - 4.20i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-3.02 + 2.47i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-3.91 - 4.77i)T + (-13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (1.06 + 5.35i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-6.32 - 1.25i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-4.60 - 11.1i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (3.39 - 6.34i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-1.07 - 0.715i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (0.135 - 0.135i)T - 97iT^{2} \) |
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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
−10.86781260141397163228446925537, −9.977156634617335220372442003049, −8.827028298777966237486674114420, −8.479282909673909058840728570214, −7.28820527770482915176973152833, −6.57866015561590302314475800636, −5.59274875209243417039033417435, −3.76871773649864428665300455146, −1.93153930920960541943154186409, −0.824854535272909170883550313312,
2.25733254215582934786030535382, 3.18586522099921540473808140782, 4.59766672509970207300661701965, 6.35029229171601436694387066888, 6.77506110941720105007080951200, 8.459868940869691549617724222572, 8.988534113198418898931706068917, 9.763112501899739232103589786320, 10.57560361083116291945602237234, 11.42589406286598766633546815448
