L(s) = 1 | − 2.46·3-s + 1.54·5-s − 4.51·7-s + 3.07·9-s − 1.39·11-s − 1.88·13-s − 3.80·15-s + 5.81·17-s − 5.42·19-s + 11.1·21-s + 1.48·23-s − 2.60·25-s − 0.173·27-s + 6.06·29-s + 2.12·31-s + 3.43·33-s − 6.97·35-s + 5.90·37-s + 4.64·39-s + 0.00128·41-s + 1.74·43-s + 4.74·45-s − 4.47·47-s + 13.3·49-s − 14.3·51-s + 1.11·53-s − 2.15·55-s + ⋯ |
L(s) = 1 | − 1.42·3-s + 0.691·5-s − 1.70·7-s + 1.02·9-s − 0.420·11-s − 0.523·13-s − 0.983·15-s + 1.41·17-s − 1.24·19-s + 2.42·21-s + 0.310·23-s − 0.521·25-s − 0.0333·27-s + 1.12·29-s + 0.381·31-s + 0.598·33-s − 1.17·35-s + 0.970·37-s + 0.744·39-s + 0.000200·41-s + 0.265·43-s + 0.707·45-s − 0.652·47-s + 1.91·49-s − 2.00·51-s + 0.153·53-s − 0.291·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 17 | \( 1 - 5.81T + 17T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 - 1.48T + 23T^{2} \) |
| 29 | \( 1 - 6.06T + 29T^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 - 0.00128T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 - 1.11T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.51294523100294702184554429368, −6.61146327128646183181682925776, −6.31699415531470185185764487879, −5.71214599038378362852812741970, −5.09643113831397826151409015312, −4.19112949579390235640701452763, −3.15636875342342952440989822504, −2.38886972433666814439220438079, −0.964951951064832553478997929232, 0,
0.964951951064832553478997929232, 2.38886972433666814439220438079, 3.15636875342342952440989822504, 4.19112949579390235640701452763, 5.09643113831397826151409015312, 5.71214599038378362852812741970, 6.31699415531470185185764487879, 6.61146327128646183181682925776, 7.51294523100294702184554429368