L(s) = 1 | + 1.41·2-s + (−2.94 + 0.581i)3-s + 2.00·4-s + (−4.16 + 0.821i)6-s + 11.4i·7-s + 2.82·8-s + (8.32 − 3.42i)9-s + 8.48i·11-s + (−5.88 + 1.16i)12-s + 10i·13-s + 16.2i·14-s + 4.00·16-s + 3.55·17-s + (11.7 − 4.83i)18-s + 10.9·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.981 + 0.193i)3-s + 0.500·4-s + (−0.693 + 0.136i)6-s + 1.64i·7-s + 0.353·8-s + (0.924 − 0.380i)9-s + 0.771i·11-s + (−0.490 + 0.0968i)12-s + 0.769i·13-s + 1.16i·14-s + 0.250·16-s + 0.209·17-s + (0.654 − 0.268i)18-s + 0.577·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23400 + 0.940133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23400 + 0.940133i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (2.94 - 0.581i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 11.4iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 3.55T + 289T^{2} \) |
| 19 | \( 1 - 10.9T + 361T^{2} \) |
| 23 | \( 1 + 17.6T + 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 8T + 961T^{2} \) |
| 37 | \( 1 + 59.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 88.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 3.55T + 2.80e3T^{2} \) |
| 59 | \( 1 - 77.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 21.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 12.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 9.02T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.688T + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.10iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 111. iT - 9.40e3T^{2} \) |
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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
−12.57824048514611523524379936269, −12.07831079669037945649985798242, −11.39581031325839089791694398442, −10.05998108095342431461236868199, −9.053513165235013559243034970168, −7.37713661315932666495708297689, −6.11638611596209040401708569206, −5.38654630373253069874280721883, −4.17313098606916738180379088757, −2.18991163283358657407662226652,
0.964200928946424616129808671689, 3.52953055754004709474721755848, 4.76969077378106107655043014792, 5.94489441369533459328623327066, 7.02528036431670267185252558277, 7.937818850961356524949436099746, 10.04973508948896779284195224586, 10.71128495996772482866525087807, 11.57404322635515980743865465586, 12.65498674180014101887927847194