| L(s) = 1 | − 3-s + 9-s + 4·11-s + 6·17-s + 4·19-s + 8·23-s − 27-s + 6·29-s − 8·31-s − 4·33-s − 10·37-s + 6·41-s + 4·43-s − 7·49-s − 6·51-s + 10·53-s − 4·57-s + 4·59-s − 2·61-s + 12·67-s − 8·69-s + 16·71-s + 2·73-s + 16·79-s + 81-s + 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.45·17-s + 0.917·19-s + 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 1.64·37-s + 0.937·41-s + 0.609·43-s − 49-s − 0.840·51-s + 1.37·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.963·69-s + 1.89·71-s + 0.234·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.514793461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.514793461\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.74189296977842, −12.60266676390723, −12.08474289197322, −11.72688516970023, −11.14789303574106, −10.81271925735716, −10.31139513774492, −9.551925833769569, −9.488372964674504, −8.881307604023878, −8.311464609440231, −7.701973069863633, −7.227514512786232, −6.737276396278512, −6.446393027160450, −5.617200400428240, −5.225949083251928, −5.006201233894822, −4.050924282510209, −3.642482158936525, −3.217004298607233, −2.446375668193335, −1.621532220807796, −1.022796720025815, −0.6820380297908390,
0.6820380297908390, 1.022796720025815, 1.621532220807796, 2.446375668193335, 3.217004298607233, 3.642482158936525, 4.050924282510209, 5.006201233894822, 5.225949083251928, 5.617200400428240, 6.446393027160450, 6.737276396278512, 7.227514512786232, 7.701973069863633, 8.311464609440231, 8.881307604023878, 9.488372964674504, 9.551925833769569, 10.31139513774492, 10.81271925735716, 11.14789303574106, 11.72688516970023, 12.08474289197322, 12.60266676390723, 12.74189296977842
