L(s) = 1 | − 2.02·2-s + 2.10·4-s + 2.65·5-s − 7-s − 0.202·8-s − 5.36·10-s + 3.06·11-s + 0.621·13-s + 2.02·14-s − 3.78·16-s + 1.54·17-s + 7.42·19-s + 5.56·20-s − 6.19·22-s + 3.41·23-s + 2.02·25-s − 1.25·26-s − 2.10·28-s − 5.77·29-s − 8.60·31-s + 8.07·32-s − 3.13·34-s − 2.65·35-s + 8.63·37-s − 15.0·38-s − 0.537·40-s + 0.418·41-s + ⋯ |
L(s) = 1 | − 1.43·2-s + 1.05·4-s + 1.18·5-s − 0.377·7-s − 0.0717·8-s − 1.69·10-s + 0.923·11-s + 0.172·13-s + 0.541·14-s − 0.947·16-s + 0.375·17-s + 1.70·19-s + 1.24·20-s − 1.32·22-s + 0.712·23-s + 0.404·25-s − 0.246·26-s − 0.396·28-s − 1.07·29-s − 1.54·31-s + 1.42·32-s − 0.538·34-s − 0.447·35-s + 1.42·37-s − 2.43·38-s − 0.0850·40-s + 0.0653·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.413415148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.413415148\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 - 0.621T + 13T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 - 8.63T + 37T^{2} \) |
| 41 | \( 1 - 0.418T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 - 3.63T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 - 8.10T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.80219390405519055514924445024, −7.29244563174451009993934583541, −6.69842026993737918799580463541, −5.79750548697182881107230383562, −5.40857709862637201296302112235, −4.20808428865246909523743052779, −3.28999166871980728704965204288, −2.30608654423330834360010739131, −1.48701834515268055038492117190, −0.806698313713398612758670245594,
0.806698313713398612758670245594, 1.48701834515268055038492117190, 2.30608654423330834360010739131, 3.28999166871980728704965204288, 4.20808428865246909523743052779, 5.40857709862637201296302112235, 5.79750548697182881107230383562, 6.69842026993737918799580463541, 7.29244563174451009993934583541, 7.80219390405519055514924445024