| L(s) = 1 | + 1.47i·2-s + i·3-s − 0.166·4-s − i·5-s − 1.47·6-s + (1.47 − 2.19i)7-s + 2.69i·8-s − 9-s + 1.47·10-s + (−1.23 + 3.07i)11-s − 0.166i·12-s + 1.98·13-s + (3.23 + 2.16i)14-s + 15-s − 4.30·16-s + 0.152·17-s + ⋯ |
| L(s) = 1 | + 1.04i·2-s + 0.577i·3-s − 0.0833·4-s − 0.447i·5-s − 0.600·6-s + (0.556 − 0.830i)7-s + 0.954i·8-s − 0.333·9-s + 0.465·10-s + (−0.372 + 0.927i)11-s − 0.0481i·12-s + 0.550·13-s + (0.864 + 0.579i)14-s + 0.258·15-s − 1.07·16-s + 0.0370·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913768598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913768598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-1.47 + 2.19i)T \) |
| 11 | \( 1 + (1.23 - 3.07i)T \) |
| good | 2 | \( 1 - 1.47iT - 2T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 0.152T + 17T^{2} \) |
| 19 | \( 1 - 6.00T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 5.77iT - 29T^{2} \) |
| 31 | \( 1 - 0.696iT - 31T^{2} \) |
| 37 | \( 1 - 0.696T + 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 + 7.53iT - 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 + 0.224T + 71T^{2} \) |
| 73 | \( 1 + 0.732T + 73T^{2} \) |
| 79 | \( 1 + 2.98iT - 79T^{2} \) |
| 83 | \( 1 + 9.97T + 83T^{2} \) |
| 89 | \( 1 + 12.0iT - 89T^{2} \) |
| 97 | \( 1 - 4.13iT - 97T^{2} \) |
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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
−9.907324279746032967117412901951, −9.190611380107402693773261198945, −8.160985179630380100221706690514, −7.62597499282839121045253970230, −6.88347201003696425136611452880, −5.80877354486016671912767296733, −4.98035922041507502766206511129, −4.39738536187285100994794981169, −3.03424399187241271760004338723, −1.44568908262504999772425635311,
0.916757727573715427590698231239, 2.15844752191180017376352628110, 2.91650136696280280136507288824, 3.83433186414497922018646741946, 5.37166758037478115815285838231, 6.05901603297016531668515959600, 7.07450588081740064114911805436, 7.924969207417653147740810452197, 8.776563019043863168427864096647, 9.638563861746948539798236744612
