| L(s) = 1 | + 1.42·2-s + 0.0371·4-s − 2.54·5-s − 7-s − 2.80·8-s − 3.62·10-s − 6.28·11-s + 2.89·13-s − 1.42·14-s − 4.07·16-s − 5.09·17-s − 4.74·19-s − 0.0944·20-s − 8.96·22-s + 5.85·23-s + 1.45·25-s + 4.13·26-s − 0.0371·28-s − 0.225·29-s + 3.19·31-s − 0.210·32-s − 7.26·34-s + 2.54·35-s − 6.42·37-s − 6.76·38-s + 7.11·40-s − 7.86·41-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.0185·4-s − 1.13·5-s − 0.377·7-s − 0.990·8-s − 1.14·10-s − 1.89·11-s + 0.803·13-s − 0.381·14-s − 1.01·16-s − 1.23·17-s − 1.08·19-s − 0.0211·20-s − 1.91·22-s + 1.22·23-s + 0.290·25-s + 0.810·26-s − 0.00702·28-s − 0.0418·29-s + 0.573·31-s − 0.0371·32-s − 1.24·34-s + 0.429·35-s − 1.05·37-s − 1.09·38-s + 1.12·40-s − 1.22·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5115616448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5115616448\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.42T + 2T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 11 | \( 1 + 6.28T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 0.225T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 4.35T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 + 3.98T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 5.28T + 79T^{2} \) |
| 83 | \( 1 + 8.77T + 83T^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 - 0.248T + 97T^{2} \) |
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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
−7.938755761843617912960501798160, −6.91578339165823889670354570228, −6.45264708136793346520483954635, −5.56865528873279498996653523647, −4.77139842934360041448413836674, −4.49691201958957807960473249002, −3.43302475899080122441394514483, −3.12668199845240845616435233940, −2.09986857656129639981128304796, −0.28197545454777865960833703298,
0.28197545454777865960833703298, 2.09986857656129639981128304796, 3.12668199845240845616435233940, 3.43302475899080122441394514483, 4.49691201958957807960473249002, 4.77139842934360041448413836674, 5.56865528873279498996653523647, 6.45264708136793346520483954635, 6.91578339165823889670354570228, 7.938755761843617912960501798160
