| L(s) = 1 | + 8.11·5-s + 9.48i·7-s − 10.5i·11-s − 20.9·13-s + 4.92·17-s + 26.9i·19-s + 43.7i·23-s + 40.9·25-s + 14.5·29-s + 25.4i·31-s + 77.0i·35-s + 12.9·37-s − 50.1·41-s − 5.03i·43-s − 40.8i·47-s + ⋯ |
| L(s) = 1 | + 1.62·5-s + 1.35i·7-s − 0.962i·11-s − 1.61·13-s + 0.289·17-s + 1.41i·19-s + 1.90i·23-s + 1.63·25-s + 0.500·29-s + 0.822i·31-s + 2.20i·35-s + 0.350·37-s − 1.22·41-s − 0.117i·43-s − 0.869i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.149807813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149807813\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8.11T + 25T^{2} \) |
| 7 | \( 1 - 9.48iT - 49T^{2} \) |
| 11 | \( 1 + 10.5iT - 121T^{2} \) |
| 13 | \( 1 + 20.9T + 169T^{2} \) |
| 17 | \( 1 - 4.92T + 289T^{2} \) |
| 19 | \( 1 - 26.9iT - 361T^{2} \) |
| 23 | \( 1 - 43.7iT - 529T^{2} \) |
| 29 | \( 1 - 14.5T + 841T^{2} \) |
| 31 | \( 1 - 25.4iT - 961T^{2} \) |
| 37 | \( 1 - 12.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 5.03iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 27.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 127.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 15.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 73.8T + 9.40e3T^{2} \) |
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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
−9.909720206401342294750415794922, −9.076398786183825351427131904340, −8.396058397816555396222761296854, −7.27538624012597350898278664588, −6.16227980712562968711765655548, −5.55157956737774701697742464229, −5.14012128738920271018321559331, −3.33691930642946646864597685247, −2.40549194627889394573290643809, −1.55094491000372837063118352376,
0.60465392801843076550749663476, 2.01690115704944984086888415461, 2.78570154666339365674711014882, 4.56182196853847327755427739686, 4.83657378419675721826131887787, 6.15862993401928748796302618507, 6.95496820580567136679191497451, 7.45826724047506549238323028835, 8.777671029169555669064956846536, 9.689271322600829939123528499357
