| L(s) = 1 | − 3·2-s − 3·3-s + 4-s + 3·5-s + 9·6-s + 21·8-s + 9·9-s − 9·10-s − 15·11-s − 3·12-s + 64·13-s − 9·15-s − 71·16-s − 84·17-s − 27·18-s + 16·19-s + 3·20-s + 45·22-s − 84·23-s − 63·24-s − 116·25-s − 192·26-s − 27·27-s − 297·29-s + 27·30-s + 253·31-s + 45·32-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.268·5-s + 0.612·6-s + 0.928·8-s + 1/3·9-s − 0.284·10-s − 0.411·11-s − 0.0721·12-s + 1.36·13-s − 0.154·15-s − 1.10·16-s − 1.19·17-s − 0.353·18-s + 0.193·19-s + 0.0335·20-s + 0.436·22-s − 0.761·23-s − 0.535·24-s − 0.927·25-s − 1.44·26-s − 0.192·27-s − 1.90·29-s + 0.164·30-s + 1.46·31-s + 0.248·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 64 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 297 T + p^{3} T^{2} \) |
| 31 | \( 1 - 253 T + p^{3} T^{2} \) |
| 37 | \( 1 + 316 T + p^{3} T^{2} \) |
| 41 | \( 1 + 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 26 T + p^{3} T^{2} \) |
| 47 | \( 1 - 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 363 T + p^{3} T^{2} \) |
| 59 | \( 1 - 15 T + p^{3} T^{2} \) |
| 61 | \( 1 - 118 T + p^{3} T^{2} \) |
| 67 | \( 1 + 370 T + p^{3} T^{2} \) |
| 71 | \( 1 + 342 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 467 T + p^{3} T^{2} \) |
| 83 | \( 1 + 477 T + p^{3} T^{2} \) |
| 89 | \( 1 + 906 T + p^{3} T^{2} \) |
| 97 | \( 1 + 503 T + p^{3} T^{2} \) |
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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
−11.74663933791515619400125166220, −10.83144801609188006745059735686, −10.04964402843276502625090531371, −8.975138931525456771633189572416, −8.079074557276253008223699212346, −6.80856713564553332987415802987, −5.56753673269928998684754335972, −4.08710906384776057019120087316, −1.70657780781485673545787325476, 0,
1.70657780781485673545787325476, 4.08710906384776057019120087316, 5.56753673269928998684754335972, 6.80856713564553332987415802987, 8.079074557276253008223699212346, 8.975138931525456771633189572416, 10.04964402843276502625090531371, 10.83144801609188006745059735686, 11.74663933791515619400125166220
