L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + 1.73i·13-s + (0.866 − 0.5i)19-s − 0.999·21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.866 + 1.5i)31-s + (1.5 − 0.866i)37-s + (0.866 − 1.49i)39-s − i·43-s + (0.499 − 0.866i)49-s − 0.999·57-s + (0.866 + 0.499i)63-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + 1.73i·13-s + (0.866 − 0.5i)19-s − 0.999·21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.866 + 1.5i)31-s + (1.5 − 0.866i)37-s + (0.866 − 1.49i)39-s − i·43-s + (0.499 − 0.866i)49-s − 0.999·57-s + (0.866 + 0.499i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9256702973\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9256702973\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.830845341724891953825556468489, −8.941838160487469573366830968034, −8.009950433078762756727340570398, −7.08447290103305448515817513961, −6.72893554145078944113405571716, −5.53351083249976991638501681293, −4.76743559030730694256277379358, −4.00273813715112323183398065451, −2.27225170434749405605762787313, −1.21422986512675298244175525548,
1.17469705874623305946668412946, 2.83329241934402238676850798534, 3.92901434078730195800013988621, 5.07275805688660808972814069861, 5.49628187751843572669988334537, 6.30014313573158314903864653018, 7.60501021934564185492426530488, 8.062189299588651202707209724559, 9.271651249613063994119844367179, 9.851780522174119798597175860201