| L(s) = 1 | + 2.71·2-s + 3-s + 5.34·4-s + 1.84·5-s + 2.71·6-s − 0.270·7-s + 9.07·8-s + 9-s + 5.00·10-s − 5.14·11-s + 5.34·12-s − 13-s − 0.733·14-s + 1.84·15-s + 13.9·16-s + 5.55·17-s + 2.71·18-s + 7.36·19-s + 9.86·20-s − 0.270·21-s − 13.9·22-s + 4.74·23-s + 9.07·24-s − 1.59·25-s − 2.71·26-s + 27-s − 1.44·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.577·3-s + 2.67·4-s + 0.824·5-s + 1.10·6-s − 0.102·7-s + 3.20·8-s + 0.333·9-s + 1.58·10-s − 1.55·11-s + 1.54·12-s − 0.277·13-s − 0.196·14-s + 0.476·15-s + 3.47·16-s + 1.34·17-s + 0.638·18-s + 1.69·19-s + 2.20·20-s − 0.0590·21-s − 2.97·22-s + 0.989·23-s + 1.85·24-s − 0.319·25-s − 0.531·26-s + 0.192·27-s − 0.273·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.445672644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.445672644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 + 0.270T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 17 | \( 1 - 5.55T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 + 8.52T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 6.51T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 8.05T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.946036175685745435112443053311, −7.55302140296305860090025785495, −6.88551154104376144492969761249, −5.80058372187262334439093157790, −5.27924479385240026975400087933, −5.02709178103717324195712091973, −3.53887407123920303286959484386, −3.26862311321575191064136973422, −2.35488034567167707044408305380, −1.57127258566467389028159432355,
1.57127258566467389028159432355, 2.35488034567167707044408305380, 3.26862311321575191064136973422, 3.53887407123920303286959484386, 5.02709178103717324195712091973, 5.27924479385240026975400087933, 5.80058372187262334439093157790, 6.88551154104376144492969761249, 7.55302140296305860090025785495, 7.946036175685745435112443053311
