| L(s) = 1 | − 0.560·2-s − 7.68·4-s + 12.9·5-s + 8.79·8-s − 7.26·10-s − 48.2·11-s − 36.3·13-s + 56.5·16-s + 83.2·17-s + 67.4·19-s − 99.5·20-s + 27.0·22-s + 30.6·23-s + 42.8·25-s + 20.3·26-s − 294.·29-s + 270.·31-s − 102.·32-s − 46.7·34-s − 204.·37-s − 37.8·38-s + 114.·40-s + 287.·41-s − 55.4·43-s + 370.·44-s − 17.1·46-s − 191.·47-s + ⋯ |
| L(s) = 1 | − 0.198·2-s − 0.960·4-s + 1.15·5-s + 0.388·8-s − 0.229·10-s − 1.32·11-s − 0.774·13-s + 0.883·16-s + 1.18·17-s + 0.814·19-s − 1.11·20-s + 0.262·22-s + 0.277·23-s + 0.342·25-s + 0.153·26-s − 1.88·29-s + 1.56·31-s − 0.564·32-s − 0.235·34-s − 0.907·37-s − 0.161·38-s + 0.450·40-s + 1.09·41-s − 0.196·43-s + 1.26·44-s − 0.0550·46-s − 0.593·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.560T + 8T^{2} \) |
| 5 | \( 1 - 12.9T + 125T^{2} \) |
| 11 | \( 1 + 48.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 204.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 55.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 191.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 521.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 381.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 155.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 65.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 256.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 318.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 77.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 836.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−9.099134005325966548476607023407, −7.921615507474618230481853505224, −7.56631522414069298780923564763, −6.15713909348922837863888638722, −5.27189916909220284715486125331, −4.98638985031370177412622639434, −3.52630624393083522607441266943, −2.49788349514918925902341456587, −1.28865727498584934176123859257, 0,
1.28865727498584934176123859257, 2.49788349514918925902341456587, 3.52630624393083522607441266943, 4.98638985031370177412622639434, 5.27189916909220284715486125331, 6.15713909348922837863888638722, 7.56631522414069298780923564763, 7.921615507474618230481853505224, 9.099134005325966548476607023407
