L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 2·11-s − 12-s + 13-s − 15-s + 16-s − 4·17-s + 18-s + 8·19-s + 20-s − 2·22-s + 3·23-s − 24-s − 4·25-s + 26-s − 27-s + 2·29-s − 30-s − 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 41 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−13.41267711729321, −13.05349710579624, −12.81079673840111, −12.04545647203710, −11.59776425954709, −11.30560990709901, −10.84033358840274, −10.23345656852583, −9.806812711567372, −9.369403908803415, −8.744136056318461, −8.110714861713692, −7.487212019071804, −7.204225813950927, −6.480809552810071, −6.133048110243061, −5.572729075155499, −4.998660893495371, −4.870625853807235, −3.996439153904761, −3.465768780085529, −2.884761426052154, −2.243867968733130, −1.597064684683571, −0.9360402924608345, 0,
0.9360402924608345, 1.597064684683571, 2.243867968733130, 2.884761426052154, 3.465768780085529, 3.996439153904761, 4.870625853807235, 4.998660893495371, 5.572729075155499, 6.133048110243061, 6.480809552810071, 7.204225813950927, 7.487212019071804, 8.110714861713692, 8.744136056318461, 9.369403908803415, 9.806812711567372, 10.23345656852583, 10.84033358840274, 11.30560990709901, 11.59776425954709, 12.04545647203710, 12.81079673840111, 13.05349710579624, 13.41267711729321