L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s − 2·15-s + 2·17-s − 4·19-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s − 4·33-s − 6·37-s − 2·39-s − 6·41-s + 4·43-s + 2·45-s − 7·49-s − 2·51-s + 2·53-s + 8·55-s + 4·57-s + 4·59-s + 2·61-s + 4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s − 49-s − 0.280·51-s + 0.274·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.192005507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.192005507\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.60302456212821382428532603865, −11.45788576598466513578332775969, −10.68009321996461187722201460512, −9.549120050570062839357860330869, −8.777512613188475551383845278968, −7.13146007277855219299925953915, −6.21040581024747445781537308376, −5.24179959934618595033608799542, −3.71523857372513194753595622108, −1.64229196717434907696557687175,
1.64229196717434907696557687175, 3.71523857372513194753595622108, 5.24179959934618595033608799542, 6.21040581024747445781537308376, 7.13146007277855219299925953915, 8.777512613188475551383845278968, 9.549120050570062839357860330869, 10.68009321996461187722201460512, 11.45788576598466513578332775969, 12.60302456212821382428532603865