| L(s) = 1 | − 0.414·2-s − 1.82·4-s − 1.82·5-s + 1.58·8-s + 0.757·10-s − 1.82·11-s − 2.82·13-s + 3·16-s − 7.65·17-s − 19-s + 3.34·20-s + 0.757·22-s − 3.82·23-s − 1.65·25-s + 1.17·26-s − 0.828·29-s − 8.82·31-s − 4.41·32-s + 3.17·34-s + 6.82·37-s + 0.414·38-s − 2.89·40-s + 3.65·41-s − 1.34·43-s + 3.34·44-s + 1.58·46-s − 11.8·47-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s − 0.817·5-s + 0.560·8-s + 0.239·10-s − 0.551·11-s − 0.784·13-s + 0.750·16-s − 1.85·17-s − 0.229·19-s + 0.747·20-s + 0.161·22-s − 0.798·23-s − 0.331·25-s + 0.229·26-s − 0.153·29-s − 1.58·31-s − 0.780·32-s + 0.543·34-s + 1.12·37-s + 0.0671·38-s − 0.458·40-s + 0.571·41-s − 0.204·43-s + 0.503·44-s + 0.233·46-s − 1.72·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04142373873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04142373873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 1.34T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 3.34T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.83263779532065215188139260766, −7.39186616028546697067545586928, −6.49441932491779124870811787795, −5.63826504828734904478674258120, −4.79828078333765794003161177689, −4.32325371252366613096075432847, −3.70221403746767035556992547579, −2.63082680657280336023198856740, −1.70140781380202816137512596075, −0.10175573901711212033374417786,
0.10175573901711212033374417786, 1.70140781380202816137512596075, 2.63082680657280336023198856740, 3.70221403746767035556992547579, 4.32325371252366613096075432847, 4.79828078333765794003161177689, 5.63826504828734904478674258120, 6.49441932491779124870811787795, 7.39186616028546697067545586928, 7.83263779532065215188139260766
