| L(s) = 1 | − 0.284·2-s − 1.91·4-s + 3.66·5-s − 7-s + 1.11·8-s − 1.04·10-s + 2.89·11-s + 1.88·13-s + 0.284·14-s + 3.52·16-s + 5.41·17-s − 4.15·19-s − 7.03·20-s − 0.822·22-s + 4.94·23-s + 8.43·25-s − 0.535·26-s + 1.91·28-s + 6.42·29-s + 8.67·31-s − 3.23·32-s − 1.54·34-s − 3.66·35-s − 7.13·37-s + 1.18·38-s + 4.08·40-s − 3.80·41-s + ⋯ |
| L(s) = 1 | − 0.201·2-s − 0.959·4-s + 1.63·5-s − 0.377·7-s + 0.394·8-s − 0.329·10-s + 0.872·11-s + 0.522·13-s + 0.0760·14-s + 0.880·16-s + 1.31·17-s − 0.953·19-s − 1.57·20-s − 0.175·22-s + 1.03·23-s + 1.68·25-s − 0.104·26-s + 0.362·28-s + 1.19·29-s + 1.55·31-s − 0.571·32-s − 0.264·34-s − 0.619·35-s − 1.17·37-s + 0.191·38-s + 0.646·40-s − 0.593·41-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
| \(L(1)\) |
\(\approx\) |
\(2.473883833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.473883833\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.284T + 2T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 6.40T + 43T^{2} \) |
| 47 | \( 1 + 6.07T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 - 7.78T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 - 0.196T + 89T^{2} \) |
| 97 | \( 1 + 4.42T + 97T^{2} \) |
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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
−8.171267525321599465108853337107, −6.85091720343761364551093092947, −6.49474435017257167412270282957, −5.70193071653552290164547155569, −5.16773246794144152487571756916, −4.36946390271617336493218399075, −3.48046865632840589428589886350, −2.67686288271129841220168907799, −1.50433157103026659238371735580, −0.916288511335393260353984549336,
0.916288511335393260353984549336, 1.50433157103026659238371735580, 2.67686288271129841220168907799, 3.48046865632840589428589886350, 4.36946390271617336493218399075, 5.16773246794144152487571756916, 5.70193071653552290164547155569, 6.49474435017257167412270282957, 6.85091720343761364551093092947, 8.171267525321599465108853337107
