| L(s) = 1 | + 2-s − 3.03·3-s + 4-s − 3.55·5-s − 3.03·6-s − 4.22·7-s + 8-s + 6.20·9-s − 3.55·10-s + 0.539·11-s − 3.03·12-s − 2.53·13-s − 4.22·14-s + 10.7·15-s + 16-s − 7.56·17-s + 6.20·18-s + 8.08·19-s − 3.55·20-s + 12.8·21-s + 0.539·22-s − 8.07·23-s − 3.03·24-s + 7.67·25-s − 2.53·26-s − 9.70·27-s − 4.22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s − 1.59·5-s − 1.23·6-s − 1.59·7-s + 0.353·8-s + 2.06·9-s − 1.12·10-s + 0.162·11-s − 0.875·12-s − 0.702·13-s − 1.12·14-s + 2.78·15-s + 0.250·16-s − 1.83·17-s + 1.46·18-s + 1.85·19-s − 0.795·20-s + 2.79·21-s + 0.115·22-s − 1.68·23-s − 0.619·24-s + 1.53·25-s − 0.497·26-s − 1.86·27-s − 0.797·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 0.539T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 + 8.07T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 - 0.736T + 41T^{2} \) |
| 43 | \( 1 - 5.40T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 0.652T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 9.44T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 9.91T + 89T^{2} \) |
| 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
−7.15542673778912405981764456319, −6.78789953067292749630717473985, −6.02380813448539092876271723670, −5.48823067700689424613154192950, −4.56047780058500263050984519303, −4.09735467242499717668003131451, −3.47197932542111482357535936114, −2.41267294637086757130592691016, −0.73908237076438028765390002149, 0,
0.73908237076438028765390002149, 2.41267294637086757130592691016, 3.47197932542111482357535936114, 4.09735467242499717668003131451, 4.56047780058500263050984519303, 5.48823067700689424613154192950, 6.02380813448539092876271723670, 6.78789953067292749630717473985, 7.15542673778912405981764456319
