| L(s) = 1 | + 4.24·2-s + 7.35·3-s + 10.0·4-s − 19.0·5-s + 31.2·6-s + 8.64·8-s + 27.1·9-s − 81.0·10-s − 13.3·11-s + 73.8·12-s + 44.5·13-s − 140.·15-s − 43.5·16-s − 44.8·17-s + 115.·18-s + 112.·19-s − 191.·20-s − 56.5·22-s + 87.0·23-s + 63.6·24-s + 238.·25-s + 189.·26-s + 0.745·27-s − 151.·29-s − 595.·30-s − 256.·31-s − 254.·32-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.41·3-s + 1.25·4-s − 1.70·5-s + 2.12·6-s + 0.382·8-s + 1.00·9-s − 2.56·10-s − 0.365·11-s + 1.77·12-s + 0.949·13-s − 2.41·15-s − 0.680·16-s − 0.639·17-s + 1.50·18-s + 1.36·19-s − 2.14·20-s − 0.548·22-s + 0.788·23-s + 0.540·24-s + 1.91·25-s + 1.42·26-s + 0.00531·27-s − 0.971·29-s − 3.62·30-s − 1.48·31-s − 1.40·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 4.24T + 8T^{2} \) |
| 3 | \( 1 - 7.35T + 27T^{2} \) |
| 5 | \( 1 + 19.0T + 125T^{2} \) |
| 11 | \( 1 + 13.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 256.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 69.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 40.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 601.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 762.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 191.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 337.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 146.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 992.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 350.T + 9.12e5T^{2} \) |
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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
−8.076746554771372023720512201374, −7.45227220624631312859474936758, −6.84091401875547171734395088086, −5.61951178059383848551068456414, −4.72940769503333424211560429607, −3.97488089656206707696328711914, −3.28907469509526877159449009757, −3.08581401739292727628990182775, −1.69803784291345373614990638541, 0,
1.69803784291345373614990638541, 3.08581401739292727628990182775, 3.28907469509526877159449009757, 3.97488089656206707696328711914, 4.72940769503333424211560429607, 5.61951178059383848551068456414, 6.84091401875547171734395088086, 7.45227220624631312859474936758, 8.076746554771372023720512201374
