| L(s) = 1 | + (5.16 − 14.7i)3-s + 48.6i·5-s + 167. i·7-s + (−189. − 151. i)9-s + 319.·11-s + 152.·13-s + (715. + 251. i)15-s − 1.32e3i·17-s − 1.61e3i·19-s + (2.46e3 + 865. i)21-s − 1.22e3·23-s + 756.·25-s + (−3.21e3 + 2.00e3i)27-s + 4.54e3i·29-s + 5.35e3i·31-s + ⋯ |
| L(s) = 1 | + (0.331 − 0.943i)3-s + 0.870i·5-s + 1.29i·7-s + (−0.780 − 0.624i)9-s + 0.795·11-s + 0.250·13-s + (0.821 + 0.288i)15-s − 1.11i·17-s − 1.02i·19-s + (1.22 + 0.428i)21-s − 0.481·23-s + 0.242·25-s + (−0.848 + 0.529i)27-s + 1.00i·29-s + 1.00i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.910581198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910581198\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.16 + 14.7i)T \) |
| good | 5 | \( 1 - 48.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 167. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 319.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 152.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.32e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.61e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.54e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.35e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 3.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.04e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.32e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 5.88e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.39e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.22e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.70e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.59e5T + 8.58e9T^{2} \) |
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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
−10.99938918352586178172009531649, −9.453964504100789092897805916311, −8.913744790865040139749440594369, −7.85951176095130398764850020522, −6.76924149821544053247642641144, −6.29214668520826038022547125292, −5.00124371840592607848461943941, −3.16604562845801870129839886048, −2.56687175337858894918640764951, −1.23934492911654307476660130006,
0.47463076493746466079829611223, 1.81728614403723644388267869002, 3.88172527880151941424664303041, 3.96519877283205364805150430554, 5.29432065527118234706806311269, 6.43113129294506529446883492258, 7.86170435966516937892512339438, 8.499405239151812802559083676734, 9.543046009142262263285099640212, 10.22473667830120029169274372005
