| L(s) = 1 | + 1.80·2-s + 1.24·4-s − 2.67·5-s + 1.66·7-s − 1.36·8-s − 4.80·10-s + 2.09·11-s + 2.30·13-s + 3.00·14-s − 4.94·16-s − 3.16·17-s + 0.855·19-s − 3.31·20-s + 3.76·22-s − 23-s + 2.13·25-s + 4.15·26-s + 2.07·28-s + 29-s − 7.33·31-s − 6.16·32-s − 5.70·34-s − 4.46·35-s + 6.32·37-s + 1.54·38-s + 3.64·40-s + 0.237·41-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.620·4-s − 1.19·5-s + 0.631·7-s − 0.482·8-s − 1.52·10-s + 0.630·11-s + 0.640·13-s + 0.803·14-s − 1.23·16-s − 0.768·17-s + 0.196·19-s − 0.741·20-s + 0.803·22-s − 0.208·23-s + 0.427·25-s + 0.815·26-s + 0.391·28-s + 0.185·29-s − 1.31·31-s − 1.08·32-s − 0.978·34-s − 0.754·35-s + 1.03·37-s + 0.249·38-s + 0.576·40-s + 0.0370·41-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 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.855T + 19T^{2} \) |
| 31 | \( 1 + 7.33T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 - 0.237T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 5.36T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 - 9.77T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 4.00T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−7.63661263249830056107664742689, −6.89774980347838417943481846911, −6.15318728958355513804001082154, −5.46749932644605229516539670152, −4.52564877743733257301560482729, −4.17992643765590452822522220490, −3.55391862037623484862795747279, −2.68907055266440254382808009750, −1.47311310753884925159083487961, 0,
1.47311310753884925159083487961, 2.68907055266440254382808009750, 3.55391862037623484862795747279, 4.17992643765590452822522220490, 4.52564877743733257301560482729, 5.46749932644605229516539670152, 6.15318728958355513804001082154, 6.89774980347838417943481846911, 7.63661263249830056107664742689
