| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 3·11-s + 4·13-s + 16-s − 2·19-s + 3·20-s + 3·22-s + 6·23-s + 4·25-s + 4·26-s − 6·29-s − 5·31-s + 32-s + 2·37-s − 2·38-s + 3·40-s − 6·41-s − 10·43-s + 3·44-s + 6·46-s + 6·47-s + 4·50-s + 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 0.904·11-s + 1.10·13-s + 1/4·16-s − 0.458·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 1.11·29-s − 0.898·31-s + 0.176·32-s + 0.328·37-s − 0.324·38-s + 0.474·40-s − 0.937·41-s − 1.52·43-s + 0.452·44-s + 0.884·46-s + 0.875·47-s + 0.565·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.133596812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.133596812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.022301181159976542679631550139, −8.153752740732516759811873175243, −6.90390849159694968636551446179, −6.51972259556560194206176883369, −5.69537931970611061608929469001, −5.14293667469015758909535382312, −4.01126047180402972127592124525, −3.26671376711039705407751493003, −2.07276435338156744028852711093, −1.33346620954041525748246641205,
1.33346620954041525748246641205, 2.07276435338156744028852711093, 3.26671376711039705407751493003, 4.01126047180402972127592124525, 5.14293667469015758909535382312, 5.69537931970611061608929469001, 6.51972259556560194206176883369, 6.90390849159694968636551446179, 8.153752740732516759811873175243, 9.022301181159976542679631550139
