| L(s) = 1 | + 5-s + 13-s − 2·19-s + 4·23-s + 25-s − 6·29-s − 6·31-s − 4·37-s − 2·41-s − 2·43-s + 2·47-s + 10·53-s − 12·59-s + 8·61-s + 65-s − 4·67-s + 4·71-s − 2·73-s + 8·79-s + 2·83-s + 10·89-s − 2·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.277·13-s − 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.07·31-s − 0.657·37-s − 0.312·41-s − 0.304·43-s + 0.291·47-s + 1.37·53-s − 1.56·59-s + 1.02·61-s + 0.124·65-s − 0.488·67-s + 0.474·71-s − 0.234·73-s + 0.900·79-s + 0.219·83-s + 1.05·89-s − 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.10704381696264, −12.77081584230898, −12.34696138398193, −11.69711148732153, −11.30782268767543, −10.78738746048623, −10.42361867524606, −9.980024634918768, −9.232808824068215, −9.047556949294351, −8.638488693354556, −7.871322893347441, −7.544795142254452, −6.891385629394199, −6.554947545487057, −5.957389136289412, −5.332601311352379, −5.173795816331409, −4.358869808536286, −3.812998991234673, −3.352552291833044, −2.688201166464483, −2.025124495871076, −1.601860930678458, −0.8111165141495813, 0,
0.8111165141495813, 1.601860930678458, 2.025124495871076, 2.688201166464483, 3.352552291833044, 3.812998991234673, 4.358869808536286, 5.173795816331409, 5.332601311352379, 5.957389136289412, 6.554947545487057, 6.891385629394199, 7.544795142254452, 7.871322893347441, 8.638488693354556, 9.047556949294351, 9.232808824068215, 9.980024634918768, 10.42361867524606, 10.78738746048623, 11.30782268767543, 11.69711148732153, 12.34696138398193, 12.77081584230898, 13.10704381696264
