| L(s) = 1 | − 2.55·2-s + 4.50·4-s + 1.64·5-s − 2.91·7-s − 6.40·8-s − 4.20·10-s + 6.56·11-s + 5.20·13-s + 7.44·14-s + 7.31·16-s + 1.25·17-s − 4.89·19-s + 7.42·20-s − 16.7·22-s + 23-s − 2.28·25-s − 13.2·26-s − 13.1·28-s + 29-s − 2.43·31-s − 5.85·32-s − 3.21·34-s − 4.80·35-s + 2.46·37-s + 12.4·38-s − 10.5·40-s + 6.73·41-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 2.25·4-s + 0.736·5-s − 1.10·7-s − 2.26·8-s − 1.32·10-s + 1.97·11-s + 1.44·13-s + 1.98·14-s + 1.82·16-s + 0.305·17-s − 1.12·19-s + 1.66·20-s − 3.56·22-s + 0.208·23-s − 0.457·25-s − 2.60·26-s − 2.48·28-s + 0.185·29-s − 0.437·31-s − 1.03·32-s − 0.550·34-s − 0.812·35-s + 0.405·37-s + 2.02·38-s − 1.66·40-s + 1.05·41-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
| \(L(1)\) |
\(\approx\) |
\(1.048603341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048603341\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 - 0.485T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 + 5.84T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−8.494438034837506849693091330418, −7.36636955280562291284947895500, −6.75002380628183884070826506290, −6.10143582997651881893784122261, −5.97361575813523207261287298141, −4.13293473673957246264448644453, −3.46775195287279639882211977542, −2.36271499052380563511290183099, −1.51572792803160518849906876088, −0.76156525780055603072790582606,
0.76156525780055603072790582606, 1.51572792803160518849906876088, 2.36271499052380563511290183099, 3.46775195287279639882211977542, 4.13293473673957246264448644453, 5.97361575813523207261287298141, 6.10143582997651881893784122261, 6.75002380628183884070826506290, 7.36636955280562291284947895500, 8.494438034837506849693091330418
