| L(s) = 1 | − 2·2-s + 4·4-s − 9·5-s − 8·8-s + 18·10-s + 57·11-s + 70·13-s + 16·16-s + 51·17-s − 5·19-s − 36·20-s − 114·22-s − 69·23-s − 44·25-s − 140·26-s − 114·29-s − 23·31-s − 32·32-s − 102·34-s − 253·37-s + 10·38-s + 72·40-s − 42·41-s − 124·43-s + 228·44-s + 138·46-s + 201·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.804·5-s − 0.353·8-s + 0.569·10-s + 1.56·11-s + 1.49·13-s + 1/4·16-s + 0.727·17-s − 0.0603·19-s − 0.402·20-s − 1.10·22-s − 0.625·23-s − 0.351·25-s − 1.05·26-s − 0.729·29-s − 0.133·31-s − 0.176·32-s − 0.514·34-s − 1.12·37-s + 0.0426·38-s + 0.284·40-s − 0.159·41-s − 0.439·43-s + 0.781·44-s + 0.442·46-s + 0.623·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.413343085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413343085\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 5 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 23 T + p^{3} T^{2} \) |
| 37 | \( 1 + 253 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 124 T + p^{3} T^{2} \) |
| 47 | \( 1 - 201 T + p^{3} T^{2} \) |
| 53 | \( 1 - 393 T + p^{3} T^{2} \) |
| 59 | \( 1 - 219 T + p^{3} T^{2} \) |
| 61 | \( 1 - 709 T + p^{3} T^{2} \) |
| 67 | \( 1 - 419 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 313 T + p^{3} T^{2} \) |
| 79 | \( 1 - 461 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1017 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1834 T + p^{3} T^{2} \) |
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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
−9.657908523701716573677501365558, −8.782318111849150115453379901192, −8.254675378988963919703068193390, −7.27861169400408350203868478452, −6.48275955181919463511981842702, −5.59211717076035799119899811109, −3.94270377596366624936137757646, −3.56898143194228684870629988144, −1.79010103899284419972114530943, −0.75901547882824325681257028093,
0.75901547882824325681257028093, 1.79010103899284419972114530943, 3.56898143194228684870629988144, 3.94270377596366624936137757646, 5.59211717076035799119899811109, 6.48275955181919463511981842702, 7.27861169400408350203868478452, 8.254675378988963919703068193390, 8.782318111849150115453379901192, 9.657908523701716573677501365558
