L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 3·14-s + 15-s + 16-s + 17-s − 18-s + 4·19-s − 20-s + 3·21-s − 22-s − 3·23-s + 24-s − 4·25-s − 27-s − 3·28-s + 7·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.566·28-s + 1.29·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.33260958661491, −12.64859407239661, −12.29096834136896, −11.93717897911209, −11.41551429670593, −11.10378375642761, −10.35735307197978, −9.944084874462316, −9.720714281370189, −9.237531874957887, −8.554808161975742, −8.055340268702380, −7.676430599233306, −7.018043652613862, −6.633280566036002, −6.196016395494266, −5.736820950263230, −5.082524505834666, −4.467487502635361, −3.823137754899048, −3.289725366512114, −2.856375197107975, −2.023030554271240, −1.304267467513087, −0.6383786065629037, 0,
0.6383786065629037, 1.304267467513087, 2.023030554271240, 2.856375197107975, 3.289725366512114, 3.823137754899048, 4.467487502635361, 5.082524505834666, 5.736820950263230, 6.196016395494266, 6.633280566036002, 7.018043652613862, 7.676430599233306, 8.055340268702380, 8.554808161975742, 9.237531874957887, 9.720714281370189, 9.944084874462316, 10.35735307197978, 11.10378375642761, 11.41551429670593, 11.93717897911209, 12.29096834136896, 12.64859407239661, 13.33260958661491