| L(s) = 1 | + (0.809 + 0.587i)2-s + (0.764 + 2.35i)3-s + (0.309 + 0.951i)4-s + (−2.23 + 0.111i)5-s + (−0.764 + 2.35i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−2.52 + 1.83i)9-s + (−1.87 − 1.22i)10-s + (1.95 + 1.41i)11-s + (−2.00 + 1.45i)12-s + (−0.582 + 0.423i)13-s + (−0.809 − 0.587i)14-s + (−1.97 − 5.17i)15-s + (−0.809 + 0.587i)16-s + (0.414 − 1.27i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.441 + 1.35i)3-s + (0.154 + 0.475i)4-s + (−0.998 + 0.0499i)5-s + (−0.312 + 0.961i)6-s − 0.377·7-s + (−0.109 + 0.336i)8-s + (−0.843 + 0.612i)9-s + (−0.592 − 0.386i)10-s + (0.588 + 0.427i)11-s + (−0.578 + 0.419i)12-s + (−0.161 + 0.117i)13-s + (−0.216 − 0.157i)14-s + (−0.508 − 1.33i)15-s + (−0.202 + 0.146i)16-s + (0.100 − 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.533233 + 1.57196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.533233 + 1.57196i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 - 0.111i)T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + (-0.764 - 2.35i)T + (-2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (-1.95 - 1.41i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.582 - 0.423i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.414 + 1.27i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.31 - 4.05i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.65 - 3.38i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.86 + 5.74i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.60 + 8.01i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.93 + 2.85i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (7.85 - 5.70i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 + (-1.06 - 3.28i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.94 - 12.1i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.85 + 1.35i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.115 - 0.0839i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.485 + 1.49i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.12 + 3.45i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.50 - 6.90i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.93 + 15.1i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.226 + 0.697i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (8.26 + 6.00i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.14 + 12.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.79734131399643143498665031756, −11.01542958814137594747688831726, −9.855278172157969133538988165149, −9.172452630342396814184146347623, −8.072539857155107002104321373655, −7.16757083275278869810346797073, −5.85796954480483160908630662727, −4.48987217264186785725992508906, −4.00932608176713005258173849792, −2.96772729830404625783721518342,
0.986537433317681608510853430986, 2.65223116769960031825119717443, 3.67671339112078566598186620860, 5.06780434348246583990085211690, 6.68728071064032037781567753659, 6.99934061406670024308782368732, 8.305430423609569617479375817054, 8.967522393170390597294154744715, 10.54080104594039333401109299726, 11.41456271082625315401856214633
