| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.847 − 2.60i)3-s + (0.309 − 0.951i)4-s + (0.769 + 2.09i)5-s + (−0.847 − 2.60i)6-s + 7-s + (−0.309 − 0.951i)8-s + (−3.65 − 2.65i)9-s + (1.85 + 1.24i)10-s + (2.19 − 1.59i)11-s + (−2.21 − 1.61i)12-s + (−3.27 − 2.37i)13-s + (0.809 − 0.587i)14-s + (6.12 − 0.226i)15-s + (−0.809 − 0.587i)16-s + (0.159 + 0.490i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.489 − 1.50i)3-s + (0.154 − 0.475i)4-s + (0.343 + 0.938i)5-s + (−0.345 − 1.06i)6-s + 0.377·7-s + (−0.109 − 0.336i)8-s + (−1.21 − 0.884i)9-s + (0.587 + 0.394i)10-s + (0.662 − 0.481i)11-s + (−0.640 − 0.465i)12-s + (−0.907 − 0.659i)13-s + (0.216 − 0.157i)14-s + (1.58 − 0.0584i)15-s + (−0.202 − 0.146i)16-s + (0.0386 + 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46481 - 1.64302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46481 - 1.64302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.769 - 2.09i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (-0.847 + 2.60i)T + (-2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (-2.19 + 1.59i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.27 + 2.37i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.159 - 0.490i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.12 - 6.52i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (5.95 - 4.32i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 1.91i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.0925 + 0.284i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.51 + 1.10i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.91 - 6.47i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.0819T + 43T^{2} \) |
| 47 | \( 1 + (1.19 - 3.66i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.08 + 12.5i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.83 - 4.96i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (10.8 - 7.90i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.05 - 3.24i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.335 + 1.03i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.41 + 5.39i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.80 + 8.64i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.97 + 9.14i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.80 + 3.49i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.18 - 15.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.67040373612526024167141982200, −10.43938251374073639684675477756, −9.538069685785205286965139286883, −8.022295357009556507115205816513, −7.48035814781687148331928427139, −6.34589102287031349702611153976, −5.66058286582534745501698220060, −3.70806674142071750434536961199, −2.59174843416248721817023087687, −1.53383127225845618815550079137,
2.43207642042770527355869535181, 4.09178264435093542886112655845, 4.61783796934121750813222671733, 5.42162645517199066338338519312, 6.89132844654101572505113114147, 8.197244662163017634864584846163, 9.181344621584117719397414607703, 9.553045103558279836578635033734, 10.72469906925195121249820828344, 11.85289821682411136142162171211
