L(s) = 1 | − 2.18i·5-s + (1 + 2.44i)7-s + 5.26i·11-s − 1.01i·13-s − 3.08i·17-s + 3.24·19-s + 0.903i·23-s + 0.242·25-s − 5.34·29-s + 5.24·31-s + (5.34 − 2.18i)35-s + 37-s + 8.35i·41-s + 4.47i·43-s − 12.8·47-s + ⋯ |
L(s) = 1 | − 0.975i·5-s + (0.377 + 0.925i)7-s + 1.58i·11-s − 0.281i·13-s − 0.748i·17-s + 0.743·19-s + 0.188i·23-s + 0.0485·25-s − 0.992·29-s + 0.941·31-s + (0.903 − 0.368i)35-s + 0.164·37-s + 1.30i·41-s + 0.682i·43-s − 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.761152755\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.761152755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 5 | \( 1 + 2.18iT - 5T^{2} \) |
| 11 | \( 1 - 5.26iT - 11T^{2} \) |
| 13 | \( 1 + 1.01iT - 13T^{2} \) |
| 17 | \( 1 + 3.08iT - 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 - 0.903iT - 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 8.35iT - 41T^{2} \) |
| 43 | \( 1 - 4.47iT - 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.88iT - 67T^{2} \) |
| 71 | \( 1 - 2.18iT - 71T^{2} \) |
| 73 | \( 1 - 14.2iT - 73T^{2} \) |
| 79 | \( 1 - 9.37iT - 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 3.98iT - 89T^{2} \) |
| 97 | \( 1 - 6.33iT - 97T^{2} \) |
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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
−8.854219838798048933805027589113, −8.099476158240777298813937654972, −7.48303079287870818494825526638, −6.58105305431216278511036863096, −5.55773829482432812511197408616, −4.91944213053861443199493135014, −4.47617055250743368603041594875, −3.10738408309340640306190959928, −2.12615392125778918629097018997, −1.16429850444188793512273569487,
0.61349949798061218896706797146, 1.89998751636591855723660198773, 3.22154397471871276366180249914, 3.59995239759796384575443723258, 4.67844512908399492092263008448, 5.70539042812354774770491534544, 6.39697177852588564856392497451, 7.10620320694043282312351886994, 7.83203043192598867815662042686, 8.500894703734447583879153836915