L(s) = 1 | + 2-s + 4-s + 8-s − 6·11-s + 13-s + 16-s + 3·17-s + 4·19-s − 6·22-s + 3·23-s + 26-s − 3·29-s − 5·31-s + 32-s + 3·34-s − 10·37-s + 4·38-s + 9·41-s − 43-s − 6·44-s + 3·46-s + 52-s − 9·53-s − 3·58-s + 9·59-s − 11·61-s − 5·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.917·19-s − 1.27·22-s + 0.625·23-s + 0.196·26-s − 0.557·29-s − 0.898·31-s + 0.176·32-s + 0.514·34-s − 1.64·37-s + 0.648·38-s + 1.40·41-s − 0.152·43-s − 0.904·44-s + 0.442·46-s + 0.138·52-s − 1.23·53-s − 0.393·58-s + 1.17·59-s − 1.40·61-s − 0.635·62-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.69202481730688, −15.35287419721318, −14.66477561388327, −14.14809752662082, −13.54943668335210, −13.20409304031670, −12.41980315328526, −12.35940865368649, −11.37539990669906, −10.82463946044759, −10.60510612147291, −9.711337106657568, −9.318305483938326, −8.328375926348129, −7.887504855275267, −7.324363668563387, −6.805307231391112, −5.833438926760018, −5.364645388869512, −5.084161696372234, −4.153665258500947, −3.356199129879531, −2.923468442585305, −2.116899308007525, −1.207526521457620, 0,
1.207526521457620, 2.116899308007525, 2.923468442585305, 3.356199129879531, 4.153665258500947, 5.084161696372234, 5.364645388869512, 5.833438926760018, 6.805307231391112, 7.324363668563387, 7.887504855275267, 8.328375926348129, 9.318305483938326, 9.711337106657568, 10.60510612147291, 10.82463946044759, 11.37539990669906, 12.35940865368649, 12.41980315328526, 13.20409304031670, 13.54943668335210, 14.14809752662082, 14.66477561388327, 15.35287419721318, 15.69202481730688