| L(s) = 1 | − 1.62·2-s − 3.23·3-s + 0.655·4-s − 0.934·5-s + 5.26·6-s − 3.14·7-s + 2.19·8-s + 7.45·9-s + 1.52·10-s − 2.12·12-s + 3.95·13-s + 5.12·14-s + 3.02·15-s − 4.88·16-s − 5.78·17-s − 12.1·18-s − 1.48·19-s − 0.612·20-s + 10.1·21-s − 1.92·23-s − 7.08·24-s − 4.12·25-s − 6.43·26-s − 14.4·27-s − 2.06·28-s − 4.13·29-s − 4.92·30-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 1.86·3-s + 0.327·4-s − 0.417·5-s + 2.15·6-s − 1.18·7-s + 0.774·8-s + 2.48·9-s + 0.481·10-s − 0.612·12-s + 1.09·13-s + 1.36·14-s + 0.779·15-s − 1.22·16-s − 1.40·17-s − 2.86·18-s − 0.340·19-s − 0.136·20-s + 2.21·21-s − 0.401·23-s − 1.44·24-s − 0.825·25-s − 1.26·26-s − 2.77·27-s − 0.389·28-s − 0.767·29-s − 0.898·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1114064521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1114064521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 0.934T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 5.59T + 71T^{2} \) |
| 73 | \( 1 + 0.0557T + 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 + 0.787T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.662973888474487575587759192460, −9.079103510463346957826070593567, −7.976585289922335353718317877082, −7.04321424263999726243825263981, −6.43259163118209955943682548165, −5.75173768126289212362004773060, −4.53699458717186230308759512366, −3.80708242198732232844287714817, −1.70753620929724999250572827392, −0.31854929753585016382867077238,
0.31854929753585016382867077238, 1.70753620929724999250572827392, 3.80708242198732232844287714817, 4.53699458717186230308759512366, 5.75173768126289212362004773060, 6.43259163118209955943682548165, 7.04321424263999726243825263981, 7.976585289922335353718317877082, 9.079103510463346957826070593567, 9.662973888474487575587759192460
