L(s) = 1 | − 2-s − 3·3-s + 4-s − 3·5-s + 3·6-s − 4·7-s − 8-s + 6·9-s + 3·10-s − 6·11-s − 3·12-s − 6·13-s + 4·14-s + 9·15-s + 16-s − 6·17-s − 6·18-s − 5·19-s − 3·20-s + 12·21-s + 6·22-s − 23-s + 3·24-s + 4·25-s + 6·26-s − 9·27-s − 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 1.51·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 1.80·11-s − 0.866·12-s − 1.66·13-s + 1.06·14-s + 2.32·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s − 1.14·19-s − 0.670·20-s + 2.61·21-s + 1.27·22-s − 0.208·23-s + 0.612·24-s + 4/5·25-s + 1.17·26-s − 1.73·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11642 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11642 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5821 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−17.22470689892625, −16.67465320618330, −16.16735340282855, −15.83709879028198, −15.23087896041019, −15.07040410816940, −13.50636739785262, −12.93519174449010, −12.34135699077555, −12.30610968238024, −11.41541318641353, −10.75782779681942, −10.56701518079405, −9.966018390070901, −9.207629260574627, −8.475300577436567, −7.495091070379581, −7.161477709691316, −6.807044431585419, −5.844011817489637, −5.350212081651227, −4.545192854519453, −3.890479016237367, −2.845222351133270, −2.031837058613338, 0, 0, 0,
2.031837058613338, 2.845222351133270, 3.890479016237367, 4.545192854519453, 5.350212081651227, 5.844011817489637, 6.807044431585419, 7.161477709691316, 7.495091070379581, 8.475300577436567, 9.207629260574627, 9.966018390070901, 10.56701518079405, 10.75782779681942, 11.41541318641353, 12.30610968238024, 12.34135699077555, 12.93519174449010, 13.50636739785262, 15.07040410816940, 15.23087896041019, 15.83709879028198, 16.16735340282855, 16.67465320618330, 17.22470689892625