L(s) = 1 | − i·2-s + (−1 + i)3-s − 4-s + (2 − i)5-s + (1 + i)6-s + (1 − i)7-s + i·8-s + i·9-s + (−1 − 2i)10-s − 2i·11-s + (1 − i)12-s + 2i·13-s + (−1 − i)14-s + (−1 + 3i)15-s + 16-s + 4·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.577 + 0.577i)3-s − 0.5·4-s + (0.894 − 0.447i)5-s + (0.408 + 0.408i)6-s + (0.377 − 0.377i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.316 − 0.632i)10-s − 0.603i·11-s + (0.288 − 0.288i)12-s + 0.554i·13-s + (−0.267 − 0.267i)14-s + (−0.258 + 0.774i)15-s + 0.250·16-s + 0.970·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20964 - 0.506267i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20964 - 0.506267i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 37 | \( 1 + (6 - i)T \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + (-5 + 5i)T - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-3 - 3i)T + 29iT^{2} \) |
| 31 | \( 1 + (-7 + 7i)T - 31iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (7 - 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1 + i)T - 59iT^{2} \) |
| 61 | \( 1 + (3 - 3i)T - 61iT^{2} \) |
| 67 | \( 1 + (3 + 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (9 - 9i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1 + i)T - 79iT^{2} \) |
| 83 | \( 1 + (5 + 5i)T + 83iT^{2} \) |
| 89 | \( 1 + (5 + 5i)T + 89iT^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−11.28381770311072609603543422432, −10.34648853603495862491516881824, −9.743408199867838337470935935711, −8.823461997305896704784924605521, −7.67713971542231896259485145227, −6.13613439725025295932208442071, −5.14830422622777510526286974851, −4.47664800244323080901804971475, −2.86806587891364778727409853097, −1.22265141455069238326682312772,
1.45751200082893960204319637478, 3.28734478069087000181489345612, 5.15287970719384896425515269860, 5.77520787672965429489541320677, 6.68526356117287462089483685564, 7.51897958150786303110733817438, 8.585083425982287523215195122926, 9.819277877016164792075999364668, 10.31116028768085837893689030303, 11.85802598157211148736903372086