| L(s) = 1 | + 5.24·2-s + 1.98·3-s + 19.5·4-s − 1.36·5-s + 10.4·6-s + 60.3·8-s − 23.0·9-s − 7.14·10-s − 18.4·11-s + 38.8·12-s − 21.2·13-s − 2.71·15-s + 160.·16-s − 126.·17-s − 120.·18-s + 50.2·19-s − 26.5·20-s − 96.8·22-s − 82.1·23-s + 120.·24-s − 123.·25-s − 111.·26-s − 99.5·27-s + 58.4·29-s − 14.2·30-s − 319.·31-s + 358.·32-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.382·3-s + 2.43·4-s − 0.121·5-s + 0.710·6-s + 2.66·8-s − 0.853·9-s − 0.225·10-s − 0.506·11-s + 0.933·12-s − 0.453·13-s − 0.0466·15-s + 2.50·16-s − 1.79·17-s − 1.58·18-s + 0.607·19-s − 0.297·20-s − 0.938·22-s − 0.745·23-s + 1.02·24-s − 0.985·25-s − 0.840·26-s − 0.709·27-s + 0.374·29-s − 0.0865·30-s − 1.85·31-s + 1.98·32-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 | \( 1 - 5.24T + 8T^{2} \) |
| 3 | \( 1 - 1.98T + 27T^{2} \) |
| 5 | \( 1 + 1.36T + 125T^{2} \) |
| 11 | \( 1 + 18.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 306.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 562.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 623.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 923.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 196.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 62.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 432.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 327.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−7.915022036354417775725006477914, −7.29884698353090426929951506210, −6.38139740747795507499058696439, −5.72406658739178305843442661126, −5.00188854963936494706009454419, −4.18181973633926177034264369626, −3.44754885407864881482281968813, −2.51910428594937161648986288011, −2.00213096680966099137219479455, 0,
2.00213096680966099137219479455, 2.51910428594937161648986288011, 3.44754885407864881482281968813, 4.18181973633926177034264369626, 5.00188854963936494706009454419, 5.72406658739178305843442661126, 6.38139740747795507499058696439, 7.29884698353090426929951506210, 7.915022036354417775725006477914
