| L(s) = 1 | + 0.791·2-s − 2.02·3-s − 1.37·4-s − 1.87·5-s − 1.60·6-s + 3.50·7-s − 2.67·8-s + 1.09·9-s − 1.48·10-s + 2.78·12-s − 2.54·13-s + 2.77·14-s + 3.79·15-s + 0.633·16-s − 7.07·17-s + 0.868·18-s − 1.64·19-s + 2.57·20-s − 7.08·21-s − 1.98·23-s + 5.40·24-s − 1.48·25-s − 2.01·26-s + 3.85·27-s − 4.80·28-s + 6.43·29-s + 3.00·30-s + ⋯ |
| L(s) = 1 | + 0.559·2-s − 1.16·3-s − 0.686·4-s − 0.838·5-s − 0.654·6-s + 1.32·7-s − 0.944·8-s + 0.365·9-s − 0.469·10-s + 0.802·12-s − 0.706·13-s + 0.740·14-s + 0.980·15-s + 0.158·16-s − 1.71·17-s + 0.204·18-s − 0.376·19-s + 0.576·20-s − 1.54·21-s − 0.413·23-s + 1.10·24-s − 0.296·25-s − 0.395·26-s + 0.741·27-s − 0.908·28-s + 1.19·29-s + 0.548·30-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
| \(L(1)\) |
\(\approx\) |
\(0.7642161482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7642161482\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.791T + 2T^{2} \) |
| 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 - 6.43T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 - 1.62T + 41T^{2} \) |
| 43 | \( 1 - 2.95T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 6.75T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 0.568T + 71T^{2} \) |
| 73 | \( 1 + 1.81T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 - 7.64T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 - 3.29T + 97T^{2} \) |
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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
−9.752516348816533698135131364427, −8.436406200627389017969353204822, −8.281948086105889283681536933998, −6.96609911335388138767733983735, −6.15728776044515133651810185242, −5.12318516526512764384671938053, −4.63251921441616440333899503770, −4.07628711716349321619554226446, −2.46604283818437685805016595164, −0.61045552442900931642182378245,
0.61045552442900931642182378245, 2.46604283818437685805016595164, 4.07628711716349321619554226446, 4.63251921441616440333899503770, 5.12318516526512764384671938053, 6.15728776044515133651810185242, 6.96609911335388138767733983735, 8.281948086105889283681536933998, 8.436406200627389017969353204822, 9.752516348816533698135131364427
