| L(s) = 1 | − 0.0958·2-s − 1.99·4-s + 1.26·5-s − 7-s + 0.382·8-s − 0.121·10-s − 3.81·11-s + 4.33·13-s + 0.0958·14-s + 3.94·16-s − 6.65·17-s + 0.635·19-s − 2.52·20-s + 0.365·22-s − 3.34·23-s − 3.39·25-s − 0.415·26-s + 1.99·28-s − 5.89·29-s − 3.22·31-s − 1.14·32-s + 0.638·34-s − 1.26·35-s + 1.02·37-s − 0.0609·38-s + 0.484·40-s + 7.10·41-s + ⋯ |
| L(s) = 1 | − 0.0677·2-s − 0.995·4-s + 0.566·5-s − 0.377·7-s + 0.135·8-s − 0.0383·10-s − 1.15·11-s + 1.20·13-s + 0.0256·14-s + 0.986·16-s − 1.61·17-s + 0.145·19-s − 0.563·20-s + 0.0779·22-s − 0.698·23-s − 0.679·25-s − 0.0814·26-s + 0.376·28-s − 1.09·29-s − 0.578·31-s − 0.202·32-s + 0.109·34-s − 0.214·35-s + 0.168·37-s − 0.00988·38-s + 0.0765·40-s + 1.11·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9598893112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9598893112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0958T + 2T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 - 0.635T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 - 1.02T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 4.51T + 43T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 6.02T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 + 7.82T + 83T^{2} \) |
| 89 | \( 1 - 1.47T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.044712530064046390697542372872, −7.18865804397161551614521422736, −6.28297393191228579917692813487, −5.70169045516614993410391342358, −5.16522028653286551396614682543, −4.15118358767886109740782869229, −3.74215182462895762514926090590, −2.60497811691639582268766851465, −1.79737321954850493243351888966, −0.48231331959614994062305457204,
0.48231331959614994062305457204, 1.79737321954850493243351888966, 2.60497811691639582268766851465, 3.74215182462895762514926090590, 4.15118358767886109740782869229, 5.16522028653286551396614682543, 5.70169045516614993410391342358, 6.28297393191228579917692813487, 7.18865804397161551614521422736, 8.044712530064046390697542372872
