L(s) = 1 | − 0.896i·5-s − i·7-s − 3.34·11-s + 4·13-s − 3.48i·17-s + i·19-s − 0.138·23-s + 4.19·25-s − 2.07i·29-s + 0.267i·31-s − 0.896·35-s − 8.26·37-s − 9.28i·41-s + 0.535i·43-s − 5.93·47-s + ⋯ |
L(s) = 1 | − 0.400i·5-s − 0.377i·7-s − 1.00·11-s + 1.10·13-s − 0.845i·17-s + 0.229i·19-s − 0.0289·23-s + 0.839·25-s − 0.384i·29-s + 0.0481i·31-s − 0.151·35-s − 1.35·37-s − 1.44i·41-s + 0.0817i·43-s − 0.865·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167965689\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167965689\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 0.896iT - 5T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3.48iT - 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 + 0.138T + 23T^{2} \) |
| 29 | \( 1 + 2.07iT - 29T^{2} \) |
| 31 | \( 1 - 0.267iT - 31T^{2} \) |
| 37 | \( 1 + 8.26T + 37T^{2} \) |
| 41 | \( 1 + 9.28iT - 41T^{2} \) |
| 43 | \( 1 - 0.535iT - 43T^{2} \) |
| 47 | \( 1 + 5.93T + 47T^{2} \) |
| 53 | \( 1 - 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 1.03T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 + 2.92iT - 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 0.535iT - 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.908154111795597726965787154950, −7.08444248696093038785889500295, −6.48723243559062464455379428916, −5.45204836721418466013882865172, −5.09549175582322548859210678425, −4.11145761865523032867797163975, −3.37687235630442993191766630102, −2.46372626430393017834904340794, −1.36855607950418212611061209078, −0.30943520918012587124233452503,
1.24025422444832665408212657608, 2.27228068855398728491646747521, 3.13926490914665287324036868599, 3.78460078281109435244200388738, 4.87930780102829960020257272760, 5.44624446160485420380927389320, 6.32888062657961872759495047141, 6.76980053249088329149143262000, 7.74663986854685002140827243848, 8.361144655753430438098598526114