L(s) = 1 | + (−1.54 − 0.501i)3-s + (2.03 + 0.923i)5-s + 2.51i·7-s + (−0.292 − 0.212i)9-s + (0.362 − 0.263i)11-s + (−3.75 + 5.17i)13-s + (−2.68 − 2.44i)15-s + (3.87 − 1.26i)17-s + (0.114 + 0.352i)19-s + (1.26 − 3.88i)21-s + (4.27 + 5.88i)23-s + (3.29 + 3.76i)25-s + (3.20 + 4.41i)27-s + (−1.82 + 5.62i)29-s + (−0.492 − 1.51i)31-s + ⋯ |
L(s) = 1 | + (−0.891 − 0.289i)3-s + (0.910 + 0.413i)5-s + 0.951i·7-s + (−0.0975 − 0.0708i)9-s + (0.109 − 0.0794i)11-s + (−1.04 + 1.43i)13-s + (−0.692 − 0.632i)15-s + (0.940 − 0.305i)17-s + (0.0262 + 0.0808i)19-s + (0.275 − 0.848i)21-s + (0.890 + 1.22i)23-s + (0.658 + 0.752i)25-s + (0.617 + 0.850i)27-s + (−0.339 + 1.04i)29-s + (−0.0885 − 0.272i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.866696 + 0.567079i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.866696 + 0.567079i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.03 - 0.923i)T \) |
good | 3 | \( 1 + (1.54 + 0.501i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 2.51iT - 7T^{2} \) |
| 11 | \( 1 + (-0.362 + 0.263i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.75 - 5.17i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.87 + 1.26i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.114 - 0.352i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.27 - 5.88i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.82 - 5.62i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.492 + 1.51i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.485 + 0.668i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.38 + 1.73i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.31iT - 43T^{2} \) |
| 47 | \( 1 + (-11.2 - 3.64i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (11.7 + 3.81i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (10.2 + 7.43i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.23 + 5.25i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.88 - 0.938i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.44 - 4.43i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (2.41 + 3.32i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.22 + 13.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.37 - 1.42i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.08 + 4.41i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.67 - 0.870i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.58440663264078315252278385931, −10.68412413624441044913176967963, −9.414245509581827223687984379440, −9.135430304785981403474325276347, −7.40542446092845383768300192783, −6.59554034643617708029034886057, −5.67416270136907201552430930643, −5.04651535162598974093020694918, −3.10114415281628972379673906238, −1.74605996657310348042962359549,
0.78287172936226206935973136512, 2.75887089816938500867726059699, 4.48335657424092342316901621525, 5.32327250041976618145855632894, 6.08575735291064309556526788824, 7.30190882285547495866080349202, 8.307769239264314145279484628518, 9.612876145728300571858707167574, 10.38606508502172128992102792510, 10.75463136112669104268647825415