| L(s) = 1 | − 9.13·2-s − 22.4·3-s + 51.4·4-s − 25·5-s + 205.·6-s + 200.·7-s − 177.·8-s + 262.·9-s + 228.·10-s + 121·11-s − 1.15e3·12-s + 280.·13-s − 1.83e3·14-s + 561.·15-s − 26.4·16-s − 1.24e3·17-s − 2.39e3·18-s − 361·19-s − 1.28e3·20-s − 4.51e3·21-s − 1.10e3·22-s + 1.18e3·23-s + 3.98e3·24-s + 625·25-s − 2.56e3·26-s − 429.·27-s + 1.03e4·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 1.44·3-s + 1.60·4-s − 0.447·5-s + 2.32·6-s + 1.54·7-s − 0.979·8-s + 1.07·9-s + 0.722·10-s + 0.301·11-s − 2.31·12-s + 0.460·13-s − 2.50·14-s + 0.644·15-s − 0.0258·16-s − 1.04·17-s − 1.74·18-s − 0.229·19-s − 0.718·20-s − 2.23·21-s − 0.486·22-s + 0.468·23-s + 1.41·24-s + 0.200·25-s − 0.743·26-s − 0.113·27-s + 2.48·28-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
| \(L(3)\) |
\(\approx\) |
\(0.6029862015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6029862015\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.13T + 32T^{2} \) |
| 3 | \( 1 + 22.4T + 243T^{2} \) |
| 7 | \( 1 - 200.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 280.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.24e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.18e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.09e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.69e4T + 8.58e9T^{2} \) |
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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
−8.893831479588033179212621246933, −8.603291738253280596337583066330, −7.53435151832536798879655671011, −6.95556833986724508083321171223, −6.00113631181942617262372020159, −4.96688629998378530143123237434, −4.19587378872467042713401395573, −2.22644984063308570138230345009, −1.21974376269394513885577069184, −0.55162975323318731062174066879,
0.55162975323318731062174066879, 1.21974376269394513885577069184, 2.22644984063308570138230345009, 4.19587378872467042713401395573, 4.96688629998378530143123237434, 6.00113631181942617262372020159, 6.95556833986724508083321171223, 7.53435151832536798879655671011, 8.603291738253280596337583066330, 8.893831479588033179212621246933
