| L(s) = 1 | + 4i·2-s − 16·4-s + 118i·7-s − 64i·8-s − 192·11-s + 1.10e3i·13-s − 472·14-s + 256·16-s + 762i·17-s + 2.74e3·19-s − 768i·22-s − 1.56e3i·23-s − 4.42e3·26-s − 1.88e3i·28-s + 5.91e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.910i·7-s − 0.353i·8-s − 0.478·11-s + 1.81i·13-s − 0.643·14-s + 0.250·16-s + 0.639i·17-s + 1.74·19-s − 0.338i·22-s − 0.617i·23-s − 1.28·26-s − 0.455i·28-s + 1.30·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.195334073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195334073\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 118iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 192T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.10e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 762iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.74e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.56e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.51e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 378T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.43e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.31e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 9.17e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.83e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.37e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.19e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.52e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.09e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.91e3iT - 8.58e9T^{2} \) |
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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
−10.76656733887670664673149642774, −9.557455293700342535705976263121, −9.004276915383647133073234122526, −8.065596996378590770652163928973, −7.05922621110082968527470876357, −6.17541032039696109384632373287, −5.24077279646477479109744971849, −4.26548087935373380214980896378, −2.86157357112046419438571410788, −1.48842805208648658581262739744,
0.31281360253494608468916559672, 1.18624832241486541605750409412, 2.83459954782321005906137828440, 3.55652948421125300300334901107, 4.92878701657058044852398638875, 5.67673421656492059811411036886, 7.33386643894994979579433650161, 7.81100620797478577951178630402, 9.061787114554650116522998804702, 10.08241192306380424153451880554
