L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 2·13-s + 16-s − 2·17-s + 4·19-s − 4·22-s − 8·23-s − 2·26-s − 6·29-s + 8·31-s + 32-s − 2·34-s + 2·37-s + 4·38-s + 2·41-s + 12·43-s − 4·44-s − 8·46-s + 8·47-s − 2·52-s + 6·53-s − 6·58-s + 4·59-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.852·22-s − 1.66·23-s − 0.392·26-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s + 0.312·41-s + 1.82·43-s − 0.603·44-s − 1.17·46-s + 1.16·47-s − 0.277·52-s + 0.824·53-s − 0.787·58-s + 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.619132225\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.619132225\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.67989661381016, −14.91351847524405, −14.44158266091788, −13.82853517901885, −13.39441484015432, −12.94208178518241, −12.23307907604422, −11.84467735035530, −11.31341743305836, −10.44990519970956, −10.27536948514556, −9.523935577364385, −8.860245399927747, −8.045448413776216, −7.537469875028804, −7.205414615665639, −6.194230042969825, −5.720189807141514, −5.263285496932080, −4.316199986806723, −4.104079401653778, −2.918812252403384, −2.585143861569479, −1.769331835916199, −0.5601283777211909,
0.5601283777211909, 1.769331835916199, 2.585143861569479, 2.918812252403384, 4.104079401653778, 4.316199986806723, 5.263285496932080, 5.720189807141514, 6.194230042969825, 7.205414615665639, 7.537469875028804, 8.045448413776216, 8.860245399927747, 9.523935577364385, 10.27536948514556, 10.44990519970956, 11.31341743305836, 11.84467735035530, 12.23307907604422, 12.94208178518241, 13.39441484015432, 13.82853517901885, 14.44158266091788, 14.91351847524405, 15.67989661381016