| L(s) = 1 | + 3.16·2-s − 3·3-s + 1.99·4-s + 0.861·5-s − 9.48·6-s − 17.4·7-s − 18.9·8-s + 9·9-s + 2.72·10-s − 11·11-s − 5.98·12-s + 53.8·13-s − 55.0·14-s − 2.58·15-s − 75.9·16-s − 101.·17-s + 28.4·18-s − 13.8·19-s + 1.71·20-s + 52.2·21-s − 34.7·22-s − 189.·23-s + 56.9·24-s − 124.·25-s + 170.·26-s − 27·27-s − 34.7·28-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 0.577·3-s + 0.249·4-s + 0.0770·5-s − 0.645·6-s − 0.940·7-s − 0.839·8-s + 0.333·9-s + 0.0860·10-s − 0.301·11-s − 0.143·12-s + 1.14·13-s − 1.05·14-s − 0.0444·15-s − 1.18·16-s − 1.44·17-s + 0.372·18-s − 0.167·19-s + 0.0192·20-s + 0.542·21-s − 0.337·22-s − 1.72·23-s + 0.484·24-s − 0.994·25-s + 1.28·26-s − 0.192·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.453965569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453965569\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 3.16T + 8T^{2} \) |
| 5 | \( 1 - 0.861T + 125T^{2} \) |
| 7 | \( 1 + 17.4T + 343T^{2} \) |
| 13 | \( 1 - 53.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 193.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 39.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 376.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 380.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 719.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 978.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 991.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 803.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 585.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 241.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 11.0T + 9.12e5T^{2} \) |
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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
−8.796301731566895888653787543383, −8.033421396838532930904926234450, −6.59999769250539988796881990245, −6.33613700623440290290345461661, −5.69825773158642055045096635115, −4.60962847558986970718871749444, −4.05538494456108656796118368188, −3.16733747402484256784348197510, −2.10370410500404604576195182221, −0.45845653413731901352279322050,
0.45845653413731901352279322050, 2.10370410500404604576195182221, 3.16733747402484256784348197510, 4.05538494456108656796118368188, 4.60962847558986970718871749444, 5.69825773158642055045096635115, 6.33613700623440290290345461661, 6.59999769250539988796881990245, 8.033421396838532930904926234450, 8.796301731566895888653787543383
