| L(s) = 1 | − 1.92·2-s + 0.0945·3-s + 1.69·4-s + 1.98·5-s − 0.181·6-s − 3.03·7-s + 0.589·8-s − 2.99·9-s − 3.81·10-s + 0.160·12-s − 5.91·13-s + 5.84·14-s + 0.187·15-s − 4.51·16-s + 2.94·17-s + 5.74·18-s − 2.71·19-s + 3.35·20-s − 0.287·21-s + 8.64·23-s + 0.0557·24-s − 1.06·25-s + 11.3·26-s − 0.566·27-s − 5.14·28-s − 0.517·29-s − 0.360·30-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.0545·3-s + 0.846·4-s + 0.886·5-s − 0.0741·6-s − 1.14·7-s + 0.208·8-s − 0.997·9-s − 1.20·10-s + 0.0462·12-s − 1.64·13-s + 1.56·14-s + 0.0484·15-s − 1.12·16-s + 0.713·17-s + 1.35·18-s − 0.623·19-s + 0.750·20-s − 0.0627·21-s + 1.80·23-s + 0.0113·24-s − 0.213·25-s + 2.22·26-s − 0.108·27-s − 0.972·28-s − 0.0961·29-s − 0.0657·30-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
| \(L(1)\) |
\(\approx\) |
\(0.6132689140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6132689140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 - 0.0945T + 3T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 8.64T + 23T^{2} \) |
| 29 | \( 1 + 0.517T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 2.52T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 - 4.95T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 4.18T + 79T^{2} \) |
| 83 | \( 1 + 8.90T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−9.475307304555251478172222594989, −9.191404835295436992723068632547, −8.130933488461358506020279803114, −7.35325123051068725216388534972, −6.51053694591055584743871209258, −5.69749735714494845535585746854, −4.63967765159942343228978344716, −2.97208938024111805564720168355, −2.30685232193464190356990307730, −0.67160754505125811821213149034,
0.67160754505125811821213149034, 2.30685232193464190356990307730, 2.97208938024111805564720168355, 4.63967765159942343228978344716, 5.69749735714494845535585746854, 6.51053694591055584743871209258, 7.35325123051068725216388534972, 8.130933488461358506020279803114, 9.191404835295436992723068632547, 9.475307304555251478172222594989
