| L(s) = 1 | − 4.68·2-s − 5.32·3-s + 13.9·4-s + 6.47·5-s + 24.9·6-s − 28.0·8-s + 1.39·9-s − 30.3·10-s + 21.1·11-s − 74.5·12-s + 31.8·13-s − 34.5·15-s + 19.7·16-s + 42.0·17-s − 6.54·18-s + 117.·19-s + 90.5·20-s − 99.1·22-s − 130.·23-s + 149.·24-s − 83.0·25-s − 149.·26-s + 136.·27-s − 278.·29-s + 161.·30-s + 199.·31-s + 131.·32-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 1.02·3-s + 1.74·4-s + 0.579·5-s + 1.70·6-s − 1.24·8-s + 0.0516·9-s − 0.960·10-s + 0.579·11-s − 1.79·12-s + 0.679·13-s − 0.594·15-s + 0.308·16-s + 0.600·17-s − 0.0857·18-s + 1.41·19-s + 1.01·20-s − 0.960·22-s − 1.18·23-s + 1.27·24-s − 0.664·25-s − 1.12·26-s + 0.972·27-s − 1.78·29-s + 0.984·30-s + 1.15·31-s + 0.729·32-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 | \( 1 + 4.68T + 8T^{2} \) |
| 3 | \( 1 + 5.32T + 27T^{2} \) |
| 5 | \( 1 - 6.47T + 125T^{2} \) |
| 11 | \( 1 - 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 208.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 100.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 591.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 136.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 658.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 220.T + 9.12e5T^{2} \) |
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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
−8.228840695935377540385643711469, −7.65687020986172650983786625126, −6.75051171486028443700464445815, −5.95973590159666295137391209281, −5.60071208875084153938194646497, −4.23498882478649124739068805540, −2.94651552507984289673345464691, −1.69450152295762759669115773021, −1.02047619884048027304793899358, 0,
1.02047619884048027304793899358, 1.69450152295762759669115773021, 2.94651552507984289673345464691, 4.23498882478649124739068805540, 5.60071208875084153938194646497, 5.95973590159666295137391209281, 6.75051171486028443700464445815, 7.65687020986172650983786625126, 8.228840695935377540385643711469
