L(s) = 1 | − i·3-s + (−2.17 + 0.539i)5-s + i·7-s − 9-s + 5.41·11-s − 4.34i·13-s + (0.539 + 2.17i)15-s + 1.07i·17-s − 4.34·19-s + 21-s − 6.34i·23-s + (4.41 − 2.34i)25-s + i·27-s + 8.83·29-s − 4.34·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.970 + 0.241i)5-s + 0.377i·7-s − 0.333·9-s + 1.63·11-s − 1.20i·13-s + (0.139 + 0.560i)15-s + 0.261i·17-s − 0.995·19-s + 0.218·21-s − 1.32i·23-s + (0.883 − 0.468i)25-s + 0.192i·27-s + 1.64·29-s − 0.779·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.970710 - 0.759037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.970710 - 0.759037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2.17 - 0.539i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 4.34iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 6.34iT - 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 + 8.68iT - 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 6.15iT - 43T^{2} \) |
| 47 | \( 1 + 6.83iT - 47T^{2} \) |
| 53 | \( 1 + 6.18iT - 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 + 0.680T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 6.49T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−10.18101122581583643800849374524, −8.718727264241038011214500505918, −8.537433700570052842278717990491, −7.38621827139592882771916611727, −6.64933764406660740135415111004, −5.84632556479999142047508828970, −4.46110931238619617887673748912, −3.57790458223595874851368343121, −2.36629133819046856623897589246, −0.68923602224855809126287836310,
1.34773931793172283447907851778, 3.21806668371009817783939205664, 4.25650789469641968859104270442, 4.54299887212047800787024304945, 6.16022267313455681960729853672, 6.91631365615558338856309102956, 7.87619455763515309666097304942, 8.942727611892243809611659591649, 9.299862808756148442222395347590, 10.40798959369547643222047473067