| L(s) = 1 | + 4·2-s − 9.45·3-s + 16·4-s − 51.4·5-s − 37.8·6-s + 212.·7-s + 64·8-s − 153.·9-s − 205.·10-s + 516.·11-s − 151.·12-s + 768.·13-s + 848.·14-s + 486.·15-s + 256·16-s + 416.·17-s − 614.·18-s + 666.·19-s − 823.·20-s − 2.00e3·21-s + 2.06e3·22-s + 1.76e3·23-s − 604.·24-s − 474.·25-s + 3.07e3·26-s + 3.74e3·27-s + 3.39e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.606·3-s + 0.5·4-s − 0.921·5-s − 0.428·6-s + 1.63·7-s + 0.353·8-s − 0.632·9-s − 0.651·10-s + 1.28·11-s − 0.303·12-s + 1.26·13-s + 1.15·14-s + 0.558·15-s + 0.250·16-s + 0.349·17-s − 0.447·18-s + 0.423·19-s − 0.460·20-s − 0.992·21-s + 0.909·22-s + 0.694·23-s − 0.214·24-s − 0.151·25-s + 0.892·26-s + 0.989·27-s + 0.818·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.365455112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365455112\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 37 | \( 1 + 1.36e3T \) |
| good | 3 | \( 1 + 9.45T + 243T^{2} \) |
| 5 | \( 1 + 51.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 212.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 516.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 768.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 416.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 666.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.31e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 7.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.94e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.59e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.10e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.80e4T + 8.58e9T^{2} \) |
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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
−13.79709169000671638470850541696, −12.10947304613703308382644506424, −11.45475417328386674999591834761, −10.99204137642191761757825579182, −8.747425546216735374904527472263, −7.66643532722303733912138570271, −6.14946059858322864683821701187, −4.88424232054739870552030777618, −3.67193675199067198256296763846, −1.27088575563181929096211336360,
1.27088575563181929096211336360, 3.67193675199067198256296763846, 4.88424232054739870552030777618, 6.14946059858322864683821701187, 7.66643532722303733912138570271, 8.747425546216735374904527472263, 10.99204137642191761757825579182, 11.45475417328386674999591834761, 12.10947304613703308382644506424, 13.79709169000671638470850541696
