L(s) = 1 | − 1.72e3·2-s + 5.90e4·3-s + 8.88e5·4-s − 1.02e8·6-s − 5.38e8·7-s + 2.08e9·8-s + 3.48e9·9-s − 6.41e10·11-s + 5.24e10·12-s + 1.30e11·13-s + 9.30e11·14-s − 5.47e12·16-s − 8.24e12·17-s − 6.02e12·18-s + 1.34e13·19-s − 3.17e13·21-s + 1.10e14·22-s + 2.33e14·23-s + 1.23e14·24-s − 2.26e14·26-s + 2.05e14·27-s − 4.78e14·28-s − 2.02e15·29-s − 6.86e15·31-s + 5.07e15·32-s − 3.78e15·33-s + 1.42e16·34-s + ⋯ |
L(s) = 1 | − 1.19·2-s + 0.577·3-s + 0.423·4-s − 0.688·6-s − 0.720·7-s + 0.687·8-s + 1/3·9-s − 0.745·11-s + 0.244·12-s + 0.263·13-s + 0.859·14-s − 1.24·16-s − 0.991·17-s − 0.397·18-s + 0.504·19-s − 0.415·21-s + 0.889·22-s + 1.17·23-s + 0.396·24-s − 0.314·26-s + 0.192·27-s − 0.305·28-s − 0.893·29-s − 1.50·31-s + 0.797·32-s − 0.430·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.7187786657\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7187786657\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p^{10} T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 27 p^{6} T + p^{21} T^{2} \) |
| 7 | \( 1 + 76918544 p T + p^{21} T^{2} \) |
| 11 | \( 1 + 64113040188 T + p^{21} T^{2} \) |
| 13 | \( 1 - 10075392922 p T + p^{21} T^{2} \) |
| 17 | \( 1 + 8242029723618 T + p^{21} T^{2} \) |
| 19 | \( 1 - 710110618580 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 233184825844776 T + p^{21} T^{2} \) |
| 29 | \( 1 + 2024562031123770 T + p^{21} T^{2} \) |
| 31 | \( 1 + 6869194988701768 T + p^{21} T^{2} \) |
| 37 | \( 1 + 3443998107027638 T + p^{21} T^{2} \) |
| 41 | \( 1 + 21842403084625158 T + p^{21} T^{2} \) |
| 43 | \( 1 - 71792816814133756 T + p^{21} T^{2} \) |
| 47 | \( 1 + 283544719418655648 T + p^{21} T^{2} \) |
| 53 | \( 1 - 2172285419049898146 T + p^{21} T^{2} \) |
| 59 | \( 1 - 1534831476719068260 T + p^{21} T^{2} \) |
| 61 | \( 1 - 4311589520797626062 T + p^{21} T^{2} \) |
| 67 | \( 1 + 9243910904037307868 T + p^{21} T^{2} \) |
| 71 | \( 1 + 20387361256404760728 T + p^{21} T^{2} \) |
| 73 | \( 1 + 16617754439328636074 T + p^{21} T^{2} \) |
| 79 | \( 1 - 67940304745507627880 T + p^{21} T^{2} \) |
| 83 | \( 1 + 39503732340682314684 T + p^{21} T^{2} \) |
| 89 | \( 1 - 41611676186839694490 T + p^{21} T^{2} \) |
| 97 | \( 1 + 57181473208903260098 T + p^{21} T^{2} \) |
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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
−10.34636485771363608691888853453, −9.343774394065293036927750667166, −8.763023026799325630342744834256, −7.63192211652867575206940536021, −6.83425875551027798180949754532, −5.23877853268689124892791676580, −3.87056402560267868688326422147, −2.67649576341684120148123580297, −1.60371098125008608318554100159, −0.41355514769802869946612605637,
0.41355514769802869946612605637, 1.60371098125008608318554100159, 2.67649576341684120148123580297, 3.87056402560267868688326422147, 5.23877853268689124892791676580, 6.83425875551027798180949754532, 7.63192211652867575206940536021, 8.763023026799325630342744834256, 9.343774394065293036927750667166, 10.34636485771363608691888853453