L(s) = 1 | − 2.89i·3-s − i·5-s − 7-s − 5.38·9-s − 2.38i·11-s − 3.40i·13-s − 2.89·15-s − 1.92·17-s − 5.79i·19-s + 2.89i·21-s + 8.76·23-s − 25-s + 6.89i·27-s − 5.86i·29-s + 1.02·31-s + ⋯ |
L(s) = 1 | − 1.67i·3-s − 0.447i·5-s − 0.377·7-s − 1.79·9-s − 0.718i·11-s − 0.945i·13-s − 0.747·15-s − 0.466·17-s − 1.32i·19-s + 0.631i·21-s + 1.82·23-s − 0.200·25-s + 1.32i·27-s − 1.08i·29-s + 0.184·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.120146129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120146129\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 2.89iT - 3T^{2} \) |
| 11 | \( 1 + 2.38iT - 11T^{2} \) |
| 13 | \( 1 + 3.40iT - 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 + 5.79iT - 19T^{2} \) |
| 23 | \( 1 - 8.76T + 23T^{2} \) |
| 29 | \( 1 + 5.86iT - 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 - 2.81iT - 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 5.40T + 47T^{2} \) |
| 53 | \( 1 - 6.97iT - 53T^{2} \) |
| 59 | \( 1 - 6.97iT - 59T^{2} \) |
| 61 | \( 1 - 14.7iT - 61T^{2} \) |
| 67 | \( 1 - 0.209iT - 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 5.79iT - 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
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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
−8.569865493089745667236457989643, −7.68598431234982025472453719674, −7.00616873672750776425758811593, −6.37245142654450568661600783919, −5.58903316374528009200562998566, −4.71393121014216437546103253741, −3.17571515127938599391635393301, −2.56502039389646904231701807476, −1.22276126376373067973429259391, −0.40992419445077651795444675953,
1.94655996479284986221289336963, 3.27688119723841179328246677484, 3.72490319127573706161840418341, 4.78551518890209605662137116457, 5.21378218837297838331471344424, 6.46287869623071431590862633452, 7.00027018009332965958423181382, 8.224783978653710534772012667133, 9.028795991731956947127092597351, 9.581969386488071493380877748011