L(s) = 1 | + (0.707 + 0.707i)2-s + (0.258 + 0.965i)3-s + 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s − 1.73i·11-s + (−0.965 + 0.258i)12-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.965 − 0.258i)18-s − i·19-s + (1.22 − 1.22i)22-s + (−0.866 − 0.500i)24-s + (−0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.258 + 0.965i)3-s + 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s − 1.73i·11-s + (−0.965 + 0.258i)12-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.965 − 0.258i)18-s − i·19-s + (1.22 − 1.22i)22-s + (−0.866 − 0.500i)24-s + (−0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.333103324\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333103324\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−11.23241101596658626293564040405, −10.35062352955300220715088587050, −9.126728597007495566547052058947, −8.500138845903439369586263444893, −7.73994378278289791267014053211, −6.34775508616017304841428482798, −5.63644179244436880119532662276, −4.69713918049670469401304889610, −3.60742329880847449449973536253, −2.88852821851012515310168268816,
1.55063002561232208260280616514, 2.55498299986526963949050251362, 3.79227514851019030460906859615, 4.99548761884097552971721869854, 5.97847321503364693208130348955, 7.02573556671341992045434765408, 7.71255398884353962794670524358, 9.071166489096610887728610664278, 9.832973523586732183179232294526, 10.69705981808287142087854012570