| L(s) = 1 | − 3·2-s + 4-s − 6·5-s + 21·8-s + 18·10-s + 57·11-s − 62·13-s − 71·16-s − 12·17-s + 124·19-s − 6·20-s − 171·22-s − 156·23-s − 89·25-s + 186·26-s − 261·29-s + 109·31-s + 45·32-s + 36·34-s + 368·37-s − 372·38-s − 126·40-s + 54·41-s + 152·43-s + 57·44-s + 468·46-s − 78·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.536·5-s + 0.928·8-s + 0.569·10-s + 1.56·11-s − 1.32·13-s − 1.10·16-s − 0.171·17-s + 1.49·19-s − 0.0670·20-s − 1.65·22-s − 1.41·23-s − 0.711·25-s + 1.40·26-s − 1.67·29-s + 0.631·31-s + 0.248·32-s + 0.181·34-s + 1.63·37-s − 1.58·38-s − 0.498·40-s + 0.205·41-s + 0.539·43-s + 0.195·44-s + 1.50·46-s − 0.242·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 12 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 156 T + p^{3} T^{2} \) |
| 29 | \( 1 + 9 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 109 T + p^{3} T^{2} \) |
| 37 | \( 1 - 368 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 78 T + p^{3} T^{2} \) |
| 53 | \( 1 + 222 T + p^{3} T^{2} \) |
| 59 | \( 1 - 285 T + p^{3} T^{2} \) |
| 61 | \( 1 - 712 T + p^{3} T^{2} \) |
| 67 | \( 1 - 170 T + p^{3} T^{2} \) |
| 71 | \( 1 + 396 T + p^{3} T^{2} \) |
| 73 | \( 1 - 475 T + p^{3} T^{2} \) |
| 79 | \( 1 + 163 T + p^{3} T^{2} \) |
| 83 | \( 1 - 27 T + p^{3} T^{2} \) |
| 89 | \( 1 - 642 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1835 T + p^{3} T^{2} \) |
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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
−9.053060552405268776591376449024, −7.910113461564911177112259977570, −7.59925007733625824799494259977, −6.69209533190761751725639748709, −5.54428107150433441202800433694, −4.38992869347861445225873788312, −3.75363983141730312778279828622, −2.20773955240398756378673438577, −1.08091419114504985085345622590, 0,
1.08091419114504985085345622590, 2.20773955240398756378673438577, 3.75363983141730312778279828622, 4.38992869347861445225873788312, 5.54428107150433441202800433694, 6.69209533190761751725639748709, 7.59925007733625824799494259977, 7.910113461564911177112259977570, 9.053060552405268776591376449024
