| L(s) = 1 | + (0.587 − 0.809i)2-s + (1.87 − 0.607i)3-s + (−0.309 − 0.951i)4-s + (0.795 − 2.08i)5-s + (0.607 − 1.87i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.700 − 0.509i)9-s + (−1.22 − 1.87i)10-s + (−1.07 − 0.782i)11-s + (−1.15 − 1.59i)12-s + (0.264 + 0.364i)13-s + (0.809 + 0.587i)14-s + (0.218 − 4.39i)15-s + (−0.809 + 0.587i)16-s + (2.96 + 0.964i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (1.07 − 0.350i)3-s + (−0.154 − 0.475i)4-s + (0.355 − 0.934i)5-s + (0.248 − 0.763i)6-s + 0.377i·7-s + (−0.336 − 0.109i)8-s + (0.233 − 0.169i)9-s + (−0.386 − 0.592i)10-s + (−0.324 − 0.236i)11-s + (−0.333 − 0.459i)12-s + (0.0734 + 0.101i)13-s + (0.216 + 0.157i)14-s + (0.0564 − 1.13i)15-s + (−0.202 + 0.146i)16-s + (0.719 + 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0888 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0888 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66467 - 1.52280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66467 - 1.52280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (-0.795 + 2.08i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (-1.87 + 0.607i)T + (2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (1.07 + 0.782i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.264 - 0.364i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.96 - 0.964i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.20 - 6.78i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.70 + 2.34i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.661 - 2.03i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 5.20i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.50 - 6.20i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.93 + 2.85i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.46iT - 43T^{2} \) |
| 47 | \( 1 + (7.81 - 2.53i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.31 + 0.751i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.65 + 4.10i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.15 + 6.64i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.38 + 1.42i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.15 - 6.62i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.45 - 2.00i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-5.08 - 15.6i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.36 + 1.74i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.72 + 2.70i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (13.6 - 4.42i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.45108084200308587857768592276, −10.20508040399396578549072291442, −9.410143245908823520874934102372, −8.435775442095052785131166990096, −7.935634879377062846877212533937, −6.18845924158911015650856002936, −5.25105622967339873520766361546, −3.94781281530424953220704965729, −2.70689344913979767356950950357, −1.56040374465339742920806614835,
2.58060321344352908012357847612, 3.38251268515284631621967483821, 4.62925573543693134078761224319, 5.96744876883918168618282734278, 7.06341307884474906469329114119, 7.78501760575879244138804471953, 8.900636709726868129854229246381, 9.701960821635536481782431820468, 10.66543472301765155921152594250, 11.66528325458067503389300496111
