| L(s) = 1 | − 2·4-s − 0.561·5-s − 7-s + 5.12·11-s + 0.561·13-s + 4·16-s − 2.43·17-s + 4·19-s + 1.12·20-s − 7.68·23-s − 4.68·25-s + 2·28-s + 4.12·29-s − 7.68·31-s + 0.561·35-s − 10.1·37-s − 9.80·41-s + 12.2·43-s − 10.2·44-s + 13.1·47-s + 49-s − 1.12·52-s + 11·53-s − 2.87·55-s + 8.56·59-s − 5.43·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s − 0.251·5-s − 0.377·7-s + 1.54·11-s + 0.155·13-s + 16-s − 0.591·17-s + 0.917·19-s + 0.251·20-s − 1.60·23-s − 0.936·25-s + 0.377·28-s + 0.765·29-s − 1.38·31-s + 0.0949·35-s − 1.66·37-s − 1.53·41-s + 1.86·43-s − 1.54·44-s + 1.91·47-s + 0.142·49-s − 0.155·52-s + 1.51·53-s − 0.387·55-s + 1.11·59-s − 0.696·61-s − 64-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 - 8.56T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.44136674505953262089157562709, −6.89606601032260173011221489390, −5.92941709857699826838269339897, −5.53460947939178620504082036143, −4.44652914758158435316360051376, −3.86119767126853105853179703567, −3.51393760001950688532530569394, −2.13895672732756781957870508733, −1.11554157380032237055958190698, 0,
1.11554157380032237055958190698, 2.13895672732756781957870508733, 3.51393760001950688532530569394, 3.86119767126853105853179703567, 4.44652914758158435316360051376, 5.53460947939178620504082036143, 5.92941709857699826838269339897, 6.89606601032260173011221489390, 7.44136674505953262089157562709
