| L(s) = 1 | + 2.24·2-s + 3.03·4-s − 2.70·5-s − 1.81·7-s + 2.33·8-s − 6.08·10-s − 1.28·11-s + 4.53·13-s − 4.07·14-s − 0.841·16-s + 4.48·17-s + 2.67·19-s − 8.23·20-s − 2.89·22-s − 23-s + 2.33·25-s + 10.1·26-s − 5.51·28-s − 29-s − 5.79·31-s − 6.55·32-s + 10.0·34-s + 4.91·35-s − 5.91·37-s + 5.99·38-s − 6.31·40-s + 7.82·41-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.51·4-s − 1.21·5-s − 0.686·7-s + 0.824·8-s − 1.92·10-s − 0.388·11-s + 1.25·13-s − 1.08·14-s − 0.210·16-s + 1.08·17-s + 0.613·19-s − 1.84·20-s − 0.616·22-s − 0.208·23-s + 0.467·25-s + 1.99·26-s − 1.04·28-s − 0.185·29-s − 1.04·31-s − 1.15·32-s + 1.72·34-s + 0.831·35-s − 0.973·37-s + 0.973·38-s − 0.999·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 - 7.87T + 59T^{2} \) |
| 61 | \( 1 - 0.0192T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 + 8.65T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 7.50T + 97T^{2} \) |
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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
−7.45553899788539319907058772688, −6.96474875045604746458085864659, −5.94784200207336705284317460634, −5.66002532850222590995188441945, −4.67743925048610530766093789282, −3.96431727160348913339256506677, −3.35354497484705821293609237626, −3.01704280031078420800866886255, −1.54108733491556030556399346268, 0,
1.54108733491556030556399346268, 3.01704280031078420800866886255, 3.35354497484705821293609237626, 3.96431727160348913339256506677, 4.67743925048610530766093789282, 5.66002532850222590995188441945, 5.94784200207336705284317460634, 6.96474875045604746458085864659, 7.45553899788539319907058772688
