| L(s) = 1 | + 2.08·2-s + 2.35·4-s + 1.43·5-s − 0.274·7-s + 0.738·8-s + 2.98·10-s − 6.10·11-s + 0.188·13-s − 0.572·14-s − 3.16·16-s − 2.24·17-s + 3.65·19-s + 3.36·20-s − 12.7·22-s + 23-s − 2.95·25-s + 0.392·26-s − 0.646·28-s + 29-s − 4.05·31-s − 8.08·32-s − 4.68·34-s − 0.392·35-s − 4.16·37-s + 7.62·38-s + 1.05·40-s − 3.24·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s + 0.639·5-s − 0.103·7-s + 0.261·8-s + 0.943·10-s − 1.84·11-s + 0.0522·13-s − 0.153·14-s − 0.791·16-s − 0.544·17-s + 0.837·19-s + 0.752·20-s − 2.71·22-s + 0.208·23-s − 0.590·25-s + 0.0770·26-s − 0.122·28-s + 0.185·29-s − 0.728·31-s − 1.42·32-s − 0.803·34-s − 0.0663·35-s − 0.685·37-s + 1.23·38-s + 0.167·40-s − 0.506·41-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 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 7 | \( 1 + 0.274T + 7T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 13 | \( 1 - 0.188T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 31 | \( 1 + 4.05T + 31T^{2} \) |
| 37 | \( 1 + 4.16T + 37T^{2} \) |
| 41 | \( 1 + 3.24T + 41T^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 2.91T + 59T^{2} \) |
| 61 | \( 1 + 2.77T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 8.81T + 79T^{2} \) |
| 83 | \( 1 + 2.42T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−7.46676438462376755762541623803, −6.84828606254146827014085211725, −5.97330518369653143482114603963, −5.40190452723561016195655699905, −5.04068145433847841974020054988, −4.13956261509266332487318391662, −3.21558287145799918246035968847, −2.63228002038097319642997300573, −1.80842188250684125179402193733, 0,
1.80842188250684125179402193733, 2.63228002038097319642997300573, 3.21558287145799918246035968847, 4.13956261509266332487318391662, 5.04068145433847841974020054988, 5.40190452723561016195655699905, 5.97330518369653143482114603963, 6.84828606254146827014085211725, 7.46676438462376755762541623803
