L(s) = 1 | − 1.76i·2-s − i·3-s − 1.09·4-s + (2.22 − 0.230i)5-s − 1.76·6-s + i·7-s − 1.58i·8-s − 9-s + (−0.405 − 3.91i)10-s − 11-s + 1.09i·12-s − 4.21i·13-s + 1.76·14-s + (−0.230 − 2.22i)15-s − 4.99·16-s − 1.43i·17-s + ⋯ |
L(s) = 1 | − 1.24i·2-s − 0.577i·3-s − 0.548·4-s + (0.994 − 0.103i)5-s − 0.718·6-s + 0.377i·7-s − 0.561i·8-s − 0.333·9-s + (−0.128 − 1.23i)10-s − 0.301·11-s + 0.316i·12-s − 1.16i·13-s + 0.470·14-s + (−0.0595 − 0.574i)15-s − 1.24·16-s − 0.347i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.896347256\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.896347256\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.22 + 0.230i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.76iT - 2T^{2} \) |
| 13 | \( 1 + 4.21iT - 13T^{2} \) |
| 17 | \( 1 + 1.43iT - 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 - 2.64iT - 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 + 0.288T + 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 - 1.62T + 41T^{2} \) |
| 43 | \( 1 + 8.21iT - 43T^{2} \) |
| 47 | \( 1 + 3.94iT - 47T^{2} \) |
| 53 | \( 1 - 5.94iT - 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 8.54iT - 73T^{2} \) |
| 79 | \( 1 - 0.690T + 79T^{2} \) |
| 83 | \( 1 + 8.67iT - 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 15.4iT - 97T^{2} \) |
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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
−9.521972544060052630537332172071, −8.991263802241034270070597844923, −7.70034487465221292460905715802, −7.01332796554752932673659511753, −5.66953247245520336812361036919, −5.36726227176918692939224923016, −3.66073750117292155547791382783, −2.75293191795838994860856354941, −1.96627760220310997605812511589, −0.823846758077685001885359055378,
1.80098394770434901621786005236, 3.13327573180963163316580726797, 4.52958998156919131669851304954, 5.24557620001733690123774533647, 6.07704958079870235587960640771, 6.75313361745431900623534605922, 7.59042075705302410991330467728, 8.470314351245572565625619109929, 9.450366927507146638574018291165, 9.779693359450130290523962340815