L(s) = 1 | + 2.70i·2-s + i·3-s − 5.34·4-s + (2.17 + 0.539i)5-s − 2.70·6-s − i·7-s − 9.04i·8-s − 9-s + (−1.46 + 5.87i)10-s + 2·11-s − 5.34i·12-s + 0.921i·13-s + 2.70·14-s + (−0.539 + 2.17i)15-s + 13.8·16-s − 1.07i·17-s + ⋯ |
L(s) = 1 | + 1.91i·2-s + 0.577i·3-s − 2.67·4-s + (0.970 + 0.241i)5-s − 1.10·6-s − 0.377i·7-s − 3.19i·8-s − 0.333·9-s + (−0.461 + 1.85i)10-s + 0.603·11-s − 1.54i·12-s + 0.255i·13-s + 0.724·14-s + (−0.139 + 0.560i)15-s + 3.45·16-s − 0.261i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.120404 + 0.983921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.120404 + 0.983921i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.17 - 0.539i)T \) |
| 7 | \( 1 + iT \) |
good | 2 | \( 1 - 2.70iT - 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.921iT - 13T^{2} \) |
| 17 | \( 1 + 1.07iT - 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 2.34iT - 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 6.52iT - 43T^{2} \) |
| 47 | \( 1 + 4.68iT - 47T^{2} \) |
| 53 | \( 1 - 3.75iT - 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 + 4.68iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 8.34T + 89T^{2} \) |
| 97 | \( 1 - 8.43iT - 97T^{2} \) |
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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
−14.30010746691742266607213787971, −13.90176400871089569727227151612, −12.70220566872127802201624935902, −10.65276762931066666301626823265, −9.484670246199797267850012101935, −8.844067668680403265045968283277, −7.35766252171978515182272696079, −6.35057381328158010672196517790, −5.35710948013680642658277880643, −4.05511280869542593063179756039,
1.51994669840817750292699618157, 2.84094034813626633848091687740, 4.65243554777495446773253593720, 6.11461818596453356629325934372, 8.398271019182241148811939884979, 9.253200390506736368513878183527, 10.25221378914337473604070958923, 11.25244514403042399458738670214, 12.38084342986702324751065534177, 12.88196609135143912413296792723