L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s − 7-s + (0.707 + 0.707i)8-s + 1.41·11-s + (−0.866 + 0.5i)13-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s − i·19-s + (−0.366 − 1.36i)22-s + (1.22 − 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 + 0.707i)26-s + (0.866 − 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s − 7-s + (0.707 + 0.707i)8-s + 1.41·11-s + (−0.866 + 0.5i)13-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s − i·19-s + (−0.366 − 1.36i)22-s + (1.22 − 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 + 0.707i)26-s + (0.866 − 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8522488781\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8522488781\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.608739714063098056867512452019, −9.189960871061941416131426413213, −8.342725265947807493791338867100, −7.17719240692743235539184392632, −6.54355512705715603549740680667, −5.29097769577975670440728418728, −4.27142567997613092875469612002, −3.44161558601796967538204356814, −2.53257321682193062792767812108, −1.09987514374400221716913900359,
1.11498877643200776361233230571, 3.09047952805879842062926080407, 3.97806517890741944639866573825, 5.16813073671530869314154585032, 5.85312411881394611447825479008, 6.85764058349519308751770156612, 7.23037395436961702916495072498, 8.285309482844904467663830194717, 9.139848980764325709547579075238, 9.764924422026460521963069800201