| L(s) = 1 | + 3-s + 9-s − 5·11-s − 2·13-s + 6·17-s + 2·19-s − 5·23-s + 27-s − 5·29-s + 4·31-s − 5·33-s − 37-s − 2·39-s + 12·41-s − 5·43-s + 2·47-s + 6·51-s − 14·53-s + 2·57-s − 2·59-s + 5·67-s − 5·69-s − 9·71-s + 10·73-s + 11·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.04·23-s + 0.192·27-s − 0.928·29-s + 0.718·31-s − 0.870·33-s − 0.164·37-s − 0.320·39-s + 1.87·41-s − 0.762·43-s + 0.291·47-s + 0.840·51-s − 1.92·53-s + 0.264·57-s − 0.260·59-s + 0.610·67-s − 0.601·69-s − 1.06·71-s + 1.17·73-s + 1.23·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.43730521782427, −14.80530782749890, −14.41553051309142, −13.83513166950991, −13.39693011926402, −12.75688526831671, −12.30644500411690, −11.87390957624709, −10.96938596988725, −10.59423805545981, −9.803444179543302, −9.708750090901406, −8.951755788753324, −8.053038830469595, −7.693646350241257, −7.623349585631492, −6.538510110774237, −5.903861152693992, −5.223758848892267, −4.830065391564886, −3.883390681123648, −3.300351880087039, −2.626466949685777, −2.063513637072999, −1.064791054107157, 0,
1.064791054107157, 2.063513637072999, 2.626466949685777, 3.300351880087039, 3.883390681123648, 4.830065391564886, 5.223758848892267, 5.903861152693992, 6.538510110774237, 7.623349585631492, 7.693646350241257, 8.053038830469595, 8.951755788753324, 9.708750090901406, 9.803444179543302, 10.59423805545981, 10.96938596988725, 11.87390957624709, 12.30644500411690, 12.75688526831671, 13.39693011926402, 13.83513166950991, 14.41553051309142, 14.80530782749890, 15.43730521782427
