| L(s) = 1 | − 2.32·2-s − 2.57·4-s + 14.1·5-s − 19.8·7-s + 24.6·8-s − 32.8·10-s + 12.7·11-s + 41.5·13-s + 46.1·14-s − 36.7·16-s + 32.9·17-s − 96.7·19-s − 36.4·20-s − 29.5·22-s − 32.1·23-s + 74.4·25-s − 96.6·26-s + 51.0·28-s + 43.8·29-s + 187.·31-s − 111.·32-s − 76.6·34-s − 279.·35-s − 411.·37-s + 225.·38-s + 347.·40-s − 152.·41-s + ⋯ |
| L(s) = 1 | − 0.823·2-s − 0.322·4-s + 1.26·5-s − 1.06·7-s + 1.08·8-s − 1.03·10-s + 0.348·11-s + 0.885·13-s + 0.880·14-s − 0.573·16-s + 0.469·17-s − 1.16·19-s − 0.407·20-s − 0.286·22-s − 0.291·23-s + 0.595·25-s − 0.728·26-s + 0.344·28-s + 0.280·29-s + 1.08·31-s − 0.616·32-s − 0.386·34-s − 1.35·35-s − 1.82·37-s + 0.961·38-s + 1.37·40-s − 0.579·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 2.32T + 8T^{2} \) |
| 5 | \( 1 - 14.1T + 125T^{2} \) |
| 7 | \( 1 + 19.8T + 343T^{2} \) |
| 11 | \( 1 - 12.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 96.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 411.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 17.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 448.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 92.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 191.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 771.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 979.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−8.693559744848770318961470186909, −7.74008949601598659327718584330, −6.58075224292903575200451446573, −6.23924727197022258790484510795, −5.28385034514967788808028362634, −4.22373704514272089546088845011, −3.26610348464607528419651855934, −2.04068662585015444420096870052, −1.16954950805397949384743383082, 0,
1.16954950805397949384743383082, 2.04068662585015444420096870052, 3.26610348464607528419651855934, 4.22373704514272089546088845011, 5.28385034514967788808028362634, 6.23924727197022258790484510795, 6.58075224292903575200451446573, 7.74008949601598659327718584330, 8.693559744848770318961470186909
