| L(s) = 1 | − 2.46·5-s + 5.48i·7-s − 10.5i·11-s + 8.96·13-s − 16.2·17-s + 2.96i·19-s − 1.46i·23-s − 18.9·25-s + 25.0·29-s − 10.5i·31-s − 13.5i·35-s − 16.9·37-s − 29.0·41-s + 34.9i·43-s − 86.1i·47-s + ⋯ |
| L(s) = 1 | − 0.492·5-s + 0.783i·7-s − 0.962i·11-s + 0.689·13-s − 0.955·17-s + 0.156i·19-s − 0.0635i·23-s − 0.757·25-s + 0.865·29-s − 0.339i·31-s − 0.385i·35-s − 0.458·37-s − 0.707·41-s + 0.813i·43-s − 1.83i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.075508418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075508418\) |
| \(L(2)\) |
|
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 | \( 1 + 2.46T + 25T^{2} \) |
| 7 | \( 1 - 5.48iT - 49T^{2} \) |
| 11 | \( 1 + 10.5iT - 121T^{2} \) |
| 13 | \( 1 - 8.96T + 169T^{2} \) |
| 17 | \( 1 + 16.2T + 289T^{2} \) |
| 19 | \( 1 - 2.96iT - 361T^{2} \) |
| 23 | \( 1 + 1.46iT - 529T^{2} \) |
| 29 | \( 1 - 25.0T + 841T^{2} \) |
| 31 | \( 1 + 10.5iT - 961T^{2} \) |
| 37 | \( 1 + 16.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 29.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 66.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 0.966T + 3.72e3T^{2} \) |
| 67 | \( 1 + 113. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 51.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 142.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 45.8T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.198078905445671762757386273634, −8.546952468558571767388041306392, −8.001283781594697721882798095327, −6.78045831318811814331785784702, −6.05707904297844424539093466483, −5.18229575254952660702198310905, −4.04293289365044354875771964397, −3.16474214210754996404059777030, −1.96197245887590614566181869230, −0.35384518659886570363726421658,
1.16275976667891159222666320054, 2.51818924908901280396714366303, 3.89984333124627846641639954128, 4.37429394545554216278109449088, 5.53305175270912188045526764029, 6.74364402709469306685101886980, 7.20393578437498884674571936828, 8.180219460527280606875082105531, 8.911809184178112187529472953953, 9.931546802143221153755303739020
