| L(s) = 1 | − 8.55·2-s + 12.0·3-s + 41.1·4-s + 25·5-s − 103.·6-s − 150.·7-s − 77.9·8-s − 97.2·9-s − 213.·10-s + 121·11-s + 496.·12-s − 71.2·13-s + 1.28e3·14-s + 301.·15-s − 649.·16-s + 933.·17-s + 831.·18-s + 361·19-s + 1.02e3·20-s − 1.81e3·21-s − 1.03e3·22-s + 2.30e3·23-s − 941.·24-s + 625·25-s + 608.·26-s − 4.10e3·27-s − 6.19e3·28-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.774·3-s + 1.28·4-s + 0.447·5-s − 1.17·6-s − 1.16·7-s − 0.430·8-s − 0.400·9-s − 0.676·10-s + 0.301·11-s + 0.995·12-s − 0.116·13-s + 1.75·14-s + 0.346·15-s − 0.633·16-s + 0.783·17-s + 0.605·18-s + 0.229·19-s + 0.574·20-s − 0.899·21-s − 0.455·22-s + 0.909·23-s − 0.333·24-s + 0.200·25-s + 0.176·26-s − 1.08·27-s − 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.052876009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.052876009\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 8.55T + 32T^{2} \) |
| 3 | \( 1 - 12.0T + 243T^{2} \) |
| 7 | \( 1 + 150.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 71.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 933.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.14e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.14e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.09e4T + 8.58e9T^{2} \) |
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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
−9.378623565940597842699647494922, −8.430559096732298377907094651263, −7.928127728860201124347996560106, −6.81721134356847836334799856461, −6.27661813594863823183759020448, −4.93247084025109100857631322051, −3.34571527806515005972567825685, −2.75542717947194785585080805712, −1.58590993345610302627581155983, −0.54578314610514960852036768868,
0.54578314610514960852036768868, 1.58590993345610302627581155983, 2.75542717947194785585080805712, 3.34571527806515005972567825685, 4.93247084025109100857631322051, 6.27661813594863823183759020448, 6.81721134356847836334799856461, 7.928127728860201124347996560106, 8.430559096732298377907094651263, 9.378623565940597842699647494922
