| L(s) = 1 | + 4.68·2-s + 3·3-s + 13.9·4-s + 19.2·5-s + 14.0·6-s + 18.4·7-s + 27.6·8-s + 9·9-s + 90.0·10-s + 11·11-s + 41.7·12-s − 26.5·13-s + 86.1·14-s + 57.7·15-s + 18.0·16-s + 82.8·17-s + 42.1·18-s − 148.·19-s + 267.·20-s + 55.2·21-s + 51.4·22-s + 86.3·23-s + 82.8·24-s + 245.·25-s − 124.·26-s + 27·27-s + 256.·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.73·4-s + 1.72·5-s + 0.955·6-s + 0.994·7-s + 1.22·8-s + 0.333·9-s + 2.84·10-s + 0.301·11-s + 1.00·12-s − 0.567·13-s + 1.64·14-s + 0.993·15-s + 0.282·16-s + 1.18·17-s + 0.551·18-s − 1.78·19-s + 2.99·20-s + 0.574·21-s + 0.498·22-s + 0.783·23-s + 0.705·24-s + 1.96·25-s − 0.938·26-s + 0.192·27-s + 1.72·28-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
| \(L(2)\) |
\(\approx\) |
\(12.33597376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.33597376\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.68T + 8T^{2} \) |
| 5 | \( 1 - 19.2T + 125T^{2} \) |
| 7 | \( 1 - 18.4T + 343T^{2} \) |
| 13 | \( 1 + 26.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 86.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 185.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 267.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 680.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 932.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 249.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 995.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 224.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 876.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 335.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 133.T + 9.12e5T^{2} \) |
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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
−8.849223784288609931018574625085, −7.896322515262080530305848214268, −6.79196683115799667347962038985, −6.29307849885721821558314660014, −5.26762591306042902895097966756, −4.98072907332605402340407096656, −3.95086354523074471474234332789, −2.87876627954091879588568810355, −2.13421859217730555063781280019, −1.46674559353214661926475077258,
1.46674559353214661926475077258, 2.13421859217730555063781280019, 2.87876627954091879588568810355, 3.95086354523074471474234332789, 4.98072907332605402340407096656, 5.26762591306042902895097966756, 6.29307849885721821558314660014, 6.79196683115799667347962038985, 7.896322515262080530305848214268, 8.849223784288609931018574625085
