| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 2·15-s + 2·19-s + 21-s + 23-s − 25-s + 27-s + 6·29-s − 6·31-s − 2·35-s − 6·37-s − 2·41-s − 2·45-s − 6·47-s + 49-s − 10·53-s + 2·57-s + 12·59-s + 2·61-s + 63-s − 8·67-s + 69-s + 4·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.516·15-s + 0.458·19-s + 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 1.37·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s + 0.120·69-s + 0.474·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.10555942276634, −15.74274816560423, −15.19122966434810, −14.60858356299189, −14.19055018182533, −13.56509180900508, −12.98199863815776, −12.28902257082646, −11.88024575898632, −11.23599608619931, −10.73269056655203, −9.994586645592473, −9.422152955207514, −8.728868051909036, −8.166945512697150, −7.794467891403980, −7.051434037027880, −6.589969506622638, −5.533048697322881, −4.956227228978669, −4.208379786229513, −3.591079860284131, −2.990612346941021, −2.051625024283689, −1.196766132841192, 0,
1.196766132841192, 2.051625024283689, 2.990612346941021, 3.591079860284131, 4.208379786229513, 4.956227228978669, 5.533048697322881, 6.589969506622638, 7.051434037027880, 7.794467891403980, 8.166945512697150, 8.728868051909036, 9.422152955207514, 9.994586645592473, 10.73269056655203, 11.23599608619931, 11.88024575898632, 12.28902257082646, 12.98199863815776, 13.56509180900508, 14.19055018182533, 14.60858356299189, 15.19122966434810, 15.74274816560423, 16.10555942276634
