| L(s) = 1 | + 2·11-s − 2·13-s − 4·17-s − 8·23-s + 2·31-s − 8·37-s + 2·41-s − 2·43-s − 10·47-s − 2·53-s + 4·59-s − 10·61-s + 2·67-s − 12·71-s − 10·73-s − 16·79-s − 16·83-s − 14·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 0.554·13-s − 0.970·17-s − 1.66·23-s + 0.359·31-s − 1.31·37-s + 0.312·41-s − 0.304·43-s − 1.45·47-s − 0.274·53-s + 0.520·59-s − 1.28·61-s + 0.244·67-s − 1.42·71-s − 1.17·73-s − 1.80·79-s − 1.75·83-s − 1.48·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.63633022190879, −13.22923491757054, −12.69497597391282, −12.12473486327329, −11.83878525284584, −11.39024349450144, −10.82260408085068, −10.27617426586131, −9.860996062951584, −9.466915056582372, −8.801581429105702, −8.446719585784534, −7.971851600230843, −7.289249274185941, −6.901159527969067, −6.398469324932067, −5.868724074222735, −5.372792006762718, −4.637189617185135, −4.256782210824661, −3.795843598590341, −2.974531916976138, −2.565940748683696, −1.648387417074006, −1.472034785642845, 0, 0,
1.472034785642845, 1.648387417074006, 2.565940748683696, 2.974531916976138, 3.795843598590341, 4.256782210824661, 4.637189617185135, 5.372792006762718, 5.868724074222735, 6.398469324932067, 6.901159527969067, 7.289249274185941, 7.971851600230843, 8.446719585784534, 8.801581429105702, 9.466915056582372, 9.860996062951584, 10.27617426586131, 10.82260408085068, 11.39024349450144, 11.83878525284584, 12.12473486327329, 12.69497597391282, 13.22923491757054, 13.63633022190879
