| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s − 11-s + 4·13-s − 3·15-s + 2·17-s − 6·19-s − 3·21-s + 5·23-s − 4·25-s − 9·27-s − 10·29-s − 31-s + 3·33-s + 35-s + 5·37-s − 12·39-s − 2·41-s − 8·43-s + 6·45-s − 8·47-s + 49-s − 6·51-s + 6·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.301·11-s + 1.10·13-s − 0.774·15-s + 0.485·17-s − 1.37·19-s − 0.654·21-s + 1.04·23-s − 4/5·25-s − 1.73·27-s − 1.85·29-s − 0.179·31-s + 0.522·33-s + 0.169·35-s + 0.821·37-s − 1.92·39-s − 0.312·41-s − 1.21·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.79069484675499405134124167303, −6.89908461616070516242753637849, −6.32502456636790018472599457014, −5.66443182968629730360532250152, −5.22476932890402599677590740058, −4.35623858040971717774555068935, −3.55867083598673558665692920578, −2.04359511557852439223184843999, −1.22377911307014756032308709056, 0,
1.22377911307014756032308709056, 2.04359511557852439223184843999, 3.55867083598673558665692920578, 4.35623858040971717774555068935, 5.22476932890402599677590740058, 5.66443182968629730360532250152, 6.32502456636790018472599457014, 6.89908461616070516242753637849, 7.79069484675499405134124167303
