L(s) = 1 | − 2i·2-s − 4·4-s − 14i·7-s + 8i·8-s − 6·11-s + 68i·13-s − 28·14-s + 16·16-s + 78i·17-s − 44·19-s + 12i·22-s − 120i·23-s + 136·26-s + 56i·28-s + 126·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s − 0.164·11-s + 1.45i·13-s − 0.534·14-s + 0.250·16-s + 1.11i·17-s − 0.531·19-s + 0.116i·22-s − 1.08i·23-s + 1.02·26-s + 0.377i·28-s + 0.806·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.303122503\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.303122503\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 14iT - 343T^{2} \) |
| 11 | \( 1 + 6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 78iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 44T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 126T + 2.43e4T^{2} \) |
| 31 | \( 1 + 244T + 2.97e4T^{2} \) |
| 37 | \( 1 - 304iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 480T + 6.89e4T^{2} \) |
| 43 | \( 1 - 104iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 600iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 258iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 534T + 2.05e5T^{2} \) |
| 61 | \( 1 - 362T + 2.26e5T^{2} \) |
| 67 | \( 1 - 268iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 972T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 396iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 972T + 7.04e5T^{2} \) |
| 97 | \( 1 - 46iT - 9.12e5T^{2} \) |
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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
−10.79055908212195378412944646315, −10.02169034745124019765302137638, −9.042088177048726839310535444920, −8.199210416913976268165464284598, −7.02497890527338251595744714774, −6.06993041983229543660622433247, −4.53450362904211865588275318406, −3.95883014500040833368426778375, −2.45923174085190700919372007706, −1.19952123479345551092621614185,
0.46752539479720959482519751673, 2.44919589353584210765304264684, 3.74001390302376578162910707036, 5.28035131384639514186325059989, 5.64816651530711963779472048450, 6.99354451089136047503674363806, 7.79970490284372622912019963914, 8.727113518480559458849071489633, 9.515525246392198693287779669934, 10.50660331137782613243107151089