L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.707 + 0.707i)7-s + 0.999i·8-s + (0.866 − 0.499i)10-s + (0.448 − 0.258i)11-s + 1.93i·13-s + (−0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)19-s + 0.999·20-s + 0.517·22-s + (1.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.965 + 1.67i)26-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.707 + 0.707i)7-s + 0.999i·8-s + (0.866 − 0.499i)10-s + (0.448 − 0.258i)11-s + 1.93i·13-s + (−0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)19-s + 0.999·20-s + 0.517·22-s + (1.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.965 + 1.67i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.047469508\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.047469508\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.93iT - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 0.517T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.105924413298151021794638876012, −8.696284024427508379840338053549, −7.53854686092347061299452697750, −6.61353873060094737500761840612, −6.22469949811418642703990589348, −5.35175532548822369586174290629, −4.56643073284846400636031092293, −3.85543577046483735629186634608, −2.67526326898411404109116665933, −1.75801393952488185007086304113,
1.09453377981973160621078244169, 2.58130300334677576549847811470, 3.12499008856637957405612568825, 3.95570015678325692651653741829, 4.97150958981125258357750665818, 5.87513499041250455681079252683, 6.51724706184028021151399160600, 7.07783203281241105752953967714, 8.028123652595972109561619834189, 9.237094599662872022299361411129