| L(s) = 1 | + 0.164·2-s + 1.43·3-s − 31.9·4-s + 0.235·6-s + 43.8·7-s − 10.5·8-s − 240.·9-s − 163.·11-s − 45.7·12-s − 430.·13-s + 7.21·14-s + 1.02e3·16-s − 740.·17-s − 39.6·18-s − 916.·19-s + 62.7·21-s − 26.8·22-s + 529·23-s − 15.0·24-s − 70.7·26-s − 692.·27-s − 1.40e3·28-s − 5.11e3·29-s − 5.70e3·31-s + 504.·32-s − 233.·33-s − 121.·34-s + ⋯ |
| L(s) = 1 | + 0.0290·2-s + 0.0917·3-s − 0.999·4-s + 0.00266·6-s + 0.338·7-s − 0.0581·8-s − 0.991·9-s − 0.406·11-s − 0.0917·12-s − 0.706·13-s + 0.00983·14-s + 0.997·16-s − 0.621·17-s − 0.0288·18-s − 0.582·19-s + 0.0310·21-s − 0.0118·22-s + 0.208·23-s − 0.00533·24-s − 0.0205·26-s − 0.182·27-s − 0.337·28-s − 1.12·29-s − 1.06·31-s + 0.0871·32-s − 0.0373·33-s − 0.0180·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7473275361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7473275361\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 0.164T + 32T^{2} \) |
| 3 | \( 1 - 1.43T + 243T^{2} \) |
| 7 | \( 1 - 43.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 163.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 430.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 740.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 916.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.40e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.19e4T + 8.58e9T^{2} \) |
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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
−9.737355523163075891827694477689, −9.072786528134809257709530979210, −8.260744552372251459738623951997, −7.51282565187335671221457620793, −6.11632671766805458541680189996, −5.22439913983841960031898923555, −4.42580118379118562015398183521, −3.26004637324374234405606251690, −2.05983678891781450257742492974, −0.40450270705452826588087993470,
0.40450270705452826588087993470, 2.05983678891781450257742492974, 3.26004637324374234405606251690, 4.42580118379118562015398183521, 5.22439913983841960031898923555, 6.11632671766805458541680189996, 7.51282565187335671221457620793, 8.260744552372251459738623951997, 9.072786528134809257709530979210, 9.737355523163075891827694477689
