| L(s) = 1 | + 3.63·2-s − 3·3-s + 5.20·4-s − 15.4·5-s − 10.9·6-s + 21.8·7-s − 10.1·8-s + 9·9-s − 56.1·10-s + 11·11-s − 15.6·12-s + 40.0·13-s + 79.2·14-s + 46.3·15-s − 78.5·16-s − 78.4·17-s + 32.7·18-s + 5.88·19-s − 80.4·20-s − 65.4·21-s + 39.9·22-s − 9.24·23-s + 30.4·24-s + 113.·25-s + 145.·26-s − 27·27-s + 113.·28-s + ⋯ |
| L(s) = 1 | + 1.28·2-s − 0.577·3-s + 0.650·4-s − 1.38·5-s − 0.741·6-s + 1.17·7-s − 0.448·8-s + 0.333·9-s − 1.77·10-s + 0.301·11-s − 0.375·12-s + 0.855·13-s + 1.51·14-s + 0.797·15-s − 1.22·16-s − 1.11·17-s + 0.428·18-s + 0.0710·19-s − 0.899·20-s − 0.680·21-s + 0.387·22-s − 0.0837·23-s + 0.259·24-s + 0.908·25-s + 1.09·26-s − 0.192·27-s + 0.766·28-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
| \(L(2)\) |
\(\approx\) |
\(2.432065266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432065266\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.63T + 8T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 - 21.8T + 343T^{2} \) |
| 13 | \( 1 - 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.88T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.24T + 1.21e4T^{2} \) |
| 29 | \( 1 + 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 254.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 156.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 171.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 222.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 526.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 886.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 966.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 348.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 742.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 103.T + 9.12e5T^{2} \) |
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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
−8.483151259505490154946106253480, −8.038071111446938745115911359898, −6.97078852834528345826202889903, −6.31717514794166788600728887055, −5.32097477970750319737505771930, −4.62118569840822313558949063318, −4.09363208910521472587190541999, −3.38322231327235686716518548411, −1.98036370847313714657407169613, −0.60398981896025152424607061428,
0.60398981896025152424607061428, 1.98036370847313714657407169613, 3.38322231327235686716518548411, 4.09363208910521472587190541999, 4.62118569840822313558949063318, 5.32097477970750319737505771930, 6.31717514794166788600728887055, 6.97078852834528345826202889903, 8.038071111446938745115911359898, 8.483151259505490154946106253480
