| L(s) = 1 | + 4·2-s + 18.0·3-s + 16·4-s − 73.3·5-s + 72.0·6-s − 227.·7-s + 64·8-s + 81.5·9-s − 293.·10-s − 371.·11-s + 288.·12-s + 804.·13-s − 908.·14-s − 1.32e3·15-s + 256·16-s + 1.52e3·17-s + 326.·18-s + 898.·19-s − 1.17e3·20-s − 4.09e3·21-s − 1.48e3·22-s + 3.09e3·23-s + 1.15e3·24-s + 2.25e3·25-s + 3.21e3·26-s − 2.90e3·27-s − 3.63e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s − 1.31·5-s + 0.817·6-s − 1.75·7-s + 0.353·8-s + 0.335·9-s − 0.927·10-s − 0.925·11-s + 0.577·12-s + 1.32·13-s − 1.23·14-s − 1.51·15-s + 0.250·16-s + 1.27·17-s + 0.237·18-s + 0.571·19-s − 0.655·20-s − 2.02·21-s − 0.654·22-s + 1.21·23-s + 0.408·24-s + 0.721·25-s + 0.933·26-s − 0.767·27-s − 0.875·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.139647004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.139647004\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 269 | \( 1 + 7.23e4T \) |
| good | 3 | \( 1 - 18.0T + 243T^{2} \) |
| 5 | \( 1 + 73.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 227.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 371.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 804.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.52e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 898.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.85e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.43e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.73e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.31e5T + 8.58e9T^{2} \) |
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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
−9.974249641259135399347826238816, −9.066612146691885752525503409302, −8.081895942980189755874320219855, −7.48434182113436005304064577504, −6.43868870350875018427569883411, −5.33523006341108514307496220094, −3.70731872162268743195706676056, −3.46389757317030150871037441468, −2.69547193146561094213993506828, −0.72999939015472210176344305559,
0.72999939015472210176344305559, 2.69547193146561094213993506828, 3.46389757317030150871037441468, 3.70731872162268743195706676056, 5.33523006341108514307496220094, 6.43868870350875018427569883411, 7.48434182113436005304064577504, 8.081895942980189755874320219855, 9.066612146691885752525503409302, 9.974249641259135399347826238816
