| L(s) = 1 | + 2·2-s − 5·3-s + 4·4-s − 5·5-s − 10·6-s + 8·8-s − 2·9-s − 10·10-s − 11-s − 20·12-s − 7·13-s + 25·15-s + 16·16-s + 51·17-s − 4·18-s − 30·19-s − 20·20-s − 2·22-s − 50·23-s − 40·24-s + 25·25-s − 14·26-s + 145·27-s + 79·29-s + 50·30-s + 212·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.962·3-s + 1/2·4-s − 0.447·5-s − 0.680·6-s + 0.353·8-s − 0.0740·9-s − 0.316·10-s − 0.0274·11-s − 0.481·12-s − 0.149·13-s + 0.430·15-s + 1/4·16-s + 0.727·17-s − 0.0523·18-s − 0.362·19-s − 0.223·20-s − 0.0193·22-s − 0.453·23-s − 0.340·24-s + 1/5·25-s − 0.105·26-s + 1.03·27-s + 0.505·29-s + 0.304·30-s + 1.22·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.784266498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784266498\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + T + p^{3} T^{2} \) |
| 13 | \( 1 + 7 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 30 T + p^{3} T^{2} \) |
| 23 | \( 1 + 50 T + p^{3} T^{2} \) |
| 29 | \( 1 - 79 T + p^{3} T^{2} \) |
| 31 | \( 1 - 212 T + p^{3} T^{2} \) |
| 37 | \( 1 + 190 T + p^{3} T^{2} \) |
| 41 | \( 1 - 308 T + p^{3} T^{2} \) |
| 43 | \( 1 - 422 T + p^{3} T^{2} \) |
| 47 | \( 1 + 121 T + p^{3} T^{2} \) |
| 53 | \( 1 - 664 T + p^{3} T^{2} \) |
| 59 | \( 1 + 628 T + p^{3} T^{2} \) |
| 61 | \( 1 - 684 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1056 T + p^{3} T^{2} \) |
| 71 | \( 1 - 744 T + p^{3} T^{2} \) |
| 73 | \( 1 + 726 T + p^{3} T^{2} \) |
| 79 | \( 1 + 407 T + p^{3} T^{2} \) |
| 83 | \( 1 + 644 T + p^{3} T^{2} \) |
| 89 | \( 1 - 880 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1351 T + p^{3} T^{2} \) |
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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
−10.77944194281250335772699263873, −10.01405394111666189071495044286, −8.621920745436298928944016970627, −7.63282277933799516550925006038, −6.59232137657742699341179605766, −5.78754104939781942689534209060, −4.92152430829587232443755231375, −3.91020183163561642837105060176, −2.60631402790608139238028349711, −0.78436153481825526182401858011,
0.78436153481825526182401858011, 2.60631402790608139238028349711, 3.91020183163561642837105060176, 4.92152430829587232443755231375, 5.78754104939781942689534209060, 6.59232137657742699341179605766, 7.63282277933799516550925006038, 8.621920745436298928944016970627, 10.01405394111666189071495044286, 10.77944194281250335772699263873
