L(s) = 1 | + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (0.324 + 1.63i)11-s + (−0.707 + 1.70i)23-s + (−0.382 − 0.923i)25-s + (−0.216 + 1.08i)29-s + (0.324 + 0.216i)37-s + (−0.382 + 0.0761i)43-s + (0.707 − 0.707i)49-s + (−0.382 − 1.92i)53-s + 63-s + (−1.92 − 0.382i)67-s + (−1.30 + 0.541i)71-s + (0.923 + 1.38i)77-s + (0.541 − 0.541i)79-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (0.324 + 1.63i)11-s + (−0.707 + 1.70i)23-s + (−0.382 − 0.923i)25-s + (−0.216 + 1.08i)29-s + (0.324 + 0.216i)37-s + (−0.382 + 0.0761i)43-s + (0.707 − 0.707i)49-s + (−0.382 − 1.92i)53-s + 63-s + (−1.92 − 0.382i)67-s + (−1.30 + 0.541i)71-s + (0.923 + 1.38i)77-s + (0.541 − 0.541i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.517539649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517539649\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.923 + 0.382i)T \) |
good | 3 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 5 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 23 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 59 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 61 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 67 | \( 1 + (1.92 + 0.382i)T + (0.923 + 0.382i)T^{2} \) |
| 71 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 83 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 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
−8.772418727159541515055104292906, −7.86637571405956608816485922922, −7.38718188423943261764261875682, −6.85778038971462878363132149714, −5.74363744227951602927858982211, −4.76117926766562735303858696327, −4.43585611507381204077067209030, −3.49157427010735364500772090721, −1.94467740070951014045742488902, −1.58594520824121668992304161100,
0.973386327739859013508760469825, 2.06052663825748978372921775620, 3.15710681451149678206240784836, 4.11781894814344171264688911636, 4.71717721193049108679084847448, 5.93218344298020509401337478245, 6.13629720267749325030479477039, 7.31727061619386046049601293630, 7.936552484361170372300379940896, 8.707183207599352653774465871315