| L(s) = 1 | − 4·2-s + 19.2·3-s + 16·4-s − 76.8·6-s + 51.1·7-s − 64·8-s + 126.·9-s + 494.·11-s + 307.·12-s + 169·13-s − 204.·14-s + 256·16-s + 2.36e3·17-s − 506.·18-s − 699.·19-s + 983.·21-s − 1.97e3·22-s + 3.80e3·23-s − 1.23e3·24-s − 676·26-s − 2.23e3·27-s + 818.·28-s + 2.26e3·29-s − 9.08e3·31-s − 1.02e3·32-s + 9.50e3·33-s − 9.45e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.23·3-s + 0.5·4-s − 0.872·6-s + 0.394·7-s − 0.353·8-s + 0.520·9-s + 1.23·11-s + 0.616·12-s + 0.277·13-s − 0.279·14-s + 0.250·16-s + 1.98·17-s − 0.368·18-s − 0.444·19-s + 0.486·21-s − 0.871·22-s + 1.50·23-s − 0.436·24-s − 0.196·26-s − 0.590·27-s + 0.197·28-s + 0.499·29-s − 1.69·31-s − 0.176·32-s + 1.51·33-s − 1.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.239594569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.239594569\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 - 19.2T + 243T^{2} \) |
| 7 | \( 1 - 51.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 494.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 2.36e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 699.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.48e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.24e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.60e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.34e4T + 8.58e9T^{2} \) |
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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
−9.386052843136875448954230395613, −9.031040630806040890852339833700, −8.125042653218812375459479426261, −7.50101419147195880763578107508, −6.49123300356450636589429254814, −5.28156226622721029818777836363, −3.78155342309024774301903675927, −3.09395232492462549755281601555, −1.82559770030333400897496479536, −0.967778186990377222386137072651,
0.967778186990377222386137072651, 1.82559770030333400897496479536, 3.09395232492462549755281601555, 3.78155342309024774301903675927, 5.28156226622721029818777836363, 6.49123300356450636589429254814, 7.50101419147195880763578107508, 8.125042653218812375459479426261, 9.031040630806040890852339833700, 9.386052843136875448954230395613
