| L(s) = 1 | + 1.56·2-s − 3-s + 0.438·4-s + 2.56·5-s − 1.56·6-s + 2.56·7-s − 2.43·8-s + 9-s + 4·10-s + 1.43·11-s − 0.438·12-s − 7.12·13-s + 4·14-s − 2.56·15-s − 4.68·16-s − 3.12·17-s + 1.56·18-s + 6·19-s + 1.12·20-s − 2.56·21-s + 2.24·22-s − 7.68·23-s + 2.43·24-s + 1.56·25-s − 11.1·26-s − 27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.219·4-s + 1.14·5-s − 0.637·6-s + 0.968·7-s − 0.862·8-s + 0.333·9-s + 1.26·10-s + 0.433·11-s − 0.126·12-s − 1.97·13-s + 1.06·14-s − 0.661·15-s − 1.17·16-s − 0.757·17-s + 0.368·18-s + 1.37·19-s + 0.251·20-s − 0.558·21-s + 0.478·22-s − 1.60·23-s + 0.497·24-s + 0.312·25-s − 2.18·26-s − 0.192·27-s + 0.212·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675930181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675930181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 3.12T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 0.876T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−13.27127701119210569103543950909, −12.13362326165327723324936322777, −11.59391194590022716442198713208, −10.05788876278658519509190524053, −9.288256657752381522305201593345, −7.53599827417415162067226879017, −6.12680278850433556848178550465, −5.26324918048344245377206094447, −4.39777182997981306072874317154, −2.29138538088782422730957685124,
2.29138538088782422730957685124, 4.39777182997981306072874317154, 5.26324918048344245377206094447, 6.12680278850433556848178550465, 7.53599827417415162067226879017, 9.288256657752381522305201593345, 10.05788876278658519509190524053, 11.59391194590022716442198713208, 12.13362326165327723324936322777, 13.27127701119210569103543950909
