L(s) = 1 | − 0.191i·3-s + i·5-s + 7-s + 2.96·9-s − 2.28i·11-s − 7.09i·13-s + 0.191·15-s − 4.56·17-s − 0.962i·19-s − 0.191i·21-s − 7.09·23-s − 25-s − 1.14i·27-s − 8.27i·29-s − 5.79·31-s + ⋯ |
L(s) = 1 | − 0.110i·3-s + 0.447i·5-s + 0.377·7-s + 0.987·9-s − 0.688i·11-s − 1.96i·13-s + 0.0495·15-s − 1.10·17-s − 0.220i·19-s − 0.0418i·21-s − 1.47·23-s − 0.200·25-s − 0.220i·27-s − 1.53i·29-s − 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.352245707\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352245707\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.191iT - 3T^{2} \) |
| 11 | \( 1 + 2.28iT - 11T^{2} \) |
| 13 | \( 1 + 7.09iT - 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 0.962iT - 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 8.27iT - 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 - 4.01iT - 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 - 8.77iT - 43T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 + 4.79iT - 53T^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 0.486iT - 61T^{2} \) |
| 67 | \( 1 - 7.20iT - 67T^{2} \) |
| 71 | \( 1 - 1.30T + 71T^{2} \) |
| 73 | \( 1 + 8.33T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 - 1.64T + 97T^{2} \) |
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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
−8.635669569261785057873992074332, −7.935040733800988139372759012050, −7.45635234516320620963718409788, −6.35860350898486456487622116929, −5.81745930743330372716938822996, −4.75558046976541487735812623844, −3.89522555788922726730240814191, −2.92073012030298943356340714115, −1.88561149345269767172552825875, −0.44708181224402386190085731154,
1.60966449444722987084514732806, 2.10144827352913669699088223214, 3.97107299339104717174965617846, 4.27608374813577039881913904805, 5.11086115242287392309006758241, 6.22665957248140430089433377666, 7.08526662545188507842875969900, 7.50300230243983926778562523522, 8.812036245097929997040554317246, 9.075478194383825700995153610277