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
−12.04784322913751460659035791405, −10.67969177647971425154510593571, −9.792640586556902561155360973352, −9.077590529140536433063937668412, −7.75622445205518795760639460638, −7.45481107778702030279106386074, −5.36507397358802565565360148860, −3.84449729807184403470562302424, −2.00232761122088763023006440406, 0,
2.00232761122088763023006440406, 3.84449729807184403470562302424, 5.36507397358802565565360148860, 7.45481107778702030279106386074, 7.75622445205518795760639460638, 9.077590529140536433063937668412, 9.792640586556902561155360973352, 10.67969177647971425154510593571, 12.04784322913751460659035791405