L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s − 4·11-s − 2i·13-s + 14-s + 16-s + 2i·17-s + 4·19-s − 4i·22-s + 8i·23-s + 2·26-s + i·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s − 1.20·11-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s + 0.917·19-s − 0.852i·22-s + 1.66i·23-s + 0.392·26-s + 0.188i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8661965648\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8661965648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.416001800746905333477650856511, −7.56772470458726805704817209972, −7.34774345659382178971157875641, −6.26936768074248212635750847128, −5.35136070724991637364620316498, −5.09125125091688346379439405377, −3.78523889567437294335980661957, −3.14175913505525784010171094206, −1.75103076499376716873229677080, −0.29039028538893551807039851917,
1.14614896765759484954306572506, 2.49762955761122481548485427500, 2.88546078775332509877633844895, 4.13987840379394228599661793466, 4.89686538493475803700876431381, 5.57725204217112317878155124369, 6.53789868662020627382767062649, 7.45755297410796835159792071799, 8.186374640050119127973168262770, 8.906932641626885430617665078138