| L(s) = 1 | + 4·2-s + 16·4-s + 96·5-s − 148·7-s + 64·8-s + 384·10-s − 384·11-s − 334·13-s − 592·14-s + 256·16-s − 576·17-s − 664·19-s + 1.53e3·20-s − 1.53e3·22-s + 3.84e3·23-s + 6.09e3·25-s − 1.33e3·26-s − 2.36e3·28-s − 96·29-s − 4.56e3·31-s + 1.02e3·32-s − 2.30e3·34-s − 1.42e4·35-s + 5.79e3·37-s − 2.65e3·38-s + 6.14e3·40-s + 6.72e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.71·5-s − 1.14·7-s + 0.353·8-s + 1.21·10-s − 0.956·11-s − 0.548·13-s − 0.807·14-s + 1/4·16-s − 0.483·17-s − 0.421·19-s + 0.858·20-s − 0.676·22-s + 1.51·23-s + 1.94·25-s − 0.387·26-s − 0.570·28-s − 0.0211·29-s − 0.852·31-s + 0.176·32-s − 0.341·34-s − 1.96·35-s + 0.696·37-s − 0.298·38-s + 0.607·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.123602411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.123602411\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 96 T + p^{5} T^{2} \) |
| 7 | \( 1 + 148 T + p^{5} T^{2} \) |
| 11 | \( 1 + 384 T + p^{5} T^{2} \) |
| 13 | \( 1 + 334 T + p^{5} T^{2} \) |
| 17 | \( 1 + 576 T + p^{5} T^{2} \) |
| 19 | \( 1 + 664 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3840 T + p^{5} T^{2} \) |
| 29 | \( 1 + 96 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4564 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5798 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6720 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14872 T + p^{5} T^{2} \) |
| 47 | \( 1 - 19200 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7776 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13056 T + p^{5} T^{2} \) |
| 61 | \( 1 - 42782 T + p^{5} T^{2} \) |
| 67 | \( 1 - 36656 T + p^{5} T^{2} \) |
| 71 | \( 1 + 64512 T + p^{5} T^{2} \) |
| 73 | \( 1 + 16810 T + p^{5} T^{2} \) |
| 79 | \( 1 - 28076 T + p^{5} T^{2} \) |
| 83 | \( 1 - 66432 T + p^{5} T^{2} \) |
| 89 | \( 1 - 81792 T + p^{5} T^{2} \) |
| 97 | \( 1 + 29938 T + p^{5} T^{2} \) |
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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
−17.49466129180326407137472810357, −16.34251531634846100275437785659, −14.82156040587080557317119381945, −13.35310454643201799098019411634, −12.86538080839562413216106029536, −10.58807490886594072782016538684, −9.402189536896673379810646710898, −6.71238824062391881201071462974, −5.37391288708458114310503425939, −2.58758120757187154637192667425,
2.58758120757187154637192667425, 5.37391288708458114310503425939, 6.71238824062391881201071462974, 9.402189536896673379810646710898, 10.58807490886594072782016538684, 12.86538080839562413216106029536, 13.35310454643201799098019411634, 14.82156040587080557317119381945, 16.34251531634846100275437785659, 17.49466129180326407137472810357
