| L(s) = 1 | + 0.751·2-s − 1.43·4-s + 4.26·5-s − 2.58·8-s + 3.20·10-s − 4.09·11-s + 2.18·13-s + 0.933·16-s + 0.590·17-s + 19-s − 6.12·20-s − 3.07·22-s + 4.36·23-s + 13.1·25-s + 1.64·26-s + 7.36·29-s − 4.51·31-s + 5.86·32-s + 0.443·34-s + 8.95·37-s + 0.751·38-s − 11.0·40-s − 2.17·41-s − 8.69·43-s + 5.88·44-s + 3.28·46-s − 11.8·47-s + ⋯ |
| L(s) = 1 | + 0.531·2-s − 0.717·4-s + 1.90·5-s − 0.912·8-s + 1.01·10-s − 1.23·11-s + 0.606·13-s + 0.233·16-s + 0.143·17-s + 0.229·19-s − 1.36·20-s − 0.656·22-s + 0.911·23-s + 2.63·25-s + 0.322·26-s + 1.36·29-s − 0.811·31-s + 1.03·32-s + 0.0759·34-s + 1.47·37-s + 0.121·38-s − 1.74·40-s − 0.340·41-s − 1.32·43-s + 0.887·44-s + 0.483·46-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
| \(L(1)\) |
\(\approx\) |
\(3.145502351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.145502351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.751T + 2T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 - 0.590T + 17T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 2.43T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 8.32T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 + 2.11T + 97T^{2} \) |
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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
−7.945566937369961987802599736359, −6.77527102693653678106616770638, −6.26224399251163397902259949126, −5.54503270858485914162207704358, −5.12294793692314386572821025923, −4.58490365123809317610550524870, −3.28776270247785876627683154656, −2.81706473932316440266081125007, −1.85386230058804575270697219773, −0.822986192185307711060082059759,
0.822986192185307711060082059759, 1.85386230058804575270697219773, 2.81706473932316440266081125007, 3.28776270247785876627683154656, 4.58490365123809317610550524870, 5.12294793692314386572821025923, 5.54503270858485914162207704358, 6.26224399251163397902259949126, 6.77527102693653678106616770638, 7.945566937369961987802599736359
