| L(s) = 1 | + 3·3-s + 5·5-s + 7·7-s + 9·9-s − 16·11-s − 62·13-s + 15·15-s − 14·17-s − 56·19-s + 21·21-s − 136·23-s + 25·25-s + 27·27-s − 154·29-s − 116·31-s − 48·33-s + 35·35-s + 6·37-s − 186·39-s − 150·41-s − 20·43-s + 45·45-s + 152·47-s + 49·49-s − 42·51-s − 78·53-s − 80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.438·11-s − 1.32·13-s + 0.258·15-s − 0.199·17-s − 0.676·19-s + 0.218·21-s − 1.23·23-s + 1/5·25-s + 0.192·27-s − 0.986·29-s − 0.672·31-s − 0.253·33-s + 0.169·35-s + 0.0266·37-s − 0.763·39-s − 0.571·41-s − 0.0709·43-s + 0.149·45-s + 0.471·47-s + 1/7·49-s − 0.115·51-s − 0.202·53-s − 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 136 T + p^{3} T^{2} \) |
| 29 | \( 1 + 154 T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 6 T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 152 T + p^{3} T^{2} \) |
| 53 | \( 1 + 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 124 T + p^{3} T^{2} \) |
| 61 | \( 1 - 166 T + p^{3} T^{2} \) |
| 67 | \( 1 - 140 T + p^{3} T^{2} \) |
| 71 | \( 1 - 204 T + p^{3} T^{2} \) |
| 73 | \( 1 + 210 T + p^{3} T^{2} \) |
| 79 | \( 1 + 984 T + p^{3} T^{2} \) |
| 83 | \( 1 - 628 T + p^{3} T^{2} \) |
| 89 | \( 1 - 138 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1202 T + p^{3} T^{2} \) |
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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
−9.456095483391263262927704748981, −8.526885133973501133856676423918, −7.72935331089389011512398398152, −6.96529627884696128397750308924, −5.79646238204255330851246926186, −4.89233999259312195463179345264, −3.88835788122713943850386442478, −2.55160390999829507558173901281, −1.82637947571073249982237504707, 0,
1.82637947571073249982237504707, 2.55160390999829507558173901281, 3.88835788122713943850386442478, 4.89233999259312195463179345264, 5.79646238204255330851246926186, 6.96529627884696128397750308924, 7.72935331089389011512398398152, 8.526885133973501133856676423918, 9.456095483391263262927704748981
