| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.900800220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900800220\) |
| \(L(2)\) |
|
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 | \( 1 + 45.8T + 9.40e3T^{2} \) |
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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
−9.452896327542130393740628957561, −8.571456473625394009095857903794, −7.78909611553801246875656716860, −6.88838501857468792450061188708, −5.95247248456836792754335194035, −5.29830765931813021413578886748, −3.95664709705934268226455568436, −3.26771843814622011404055368035, −1.78854193779582947590308629225, −0.58499728805247873210903895453,
1.41673740643647942939761569129, 2.41469028844723940168996014708, 3.59198136158319168432289017025, 4.72147962638414181877291534338, 5.72277999686866571490641759685, 6.23470396329844666692782744259, 7.45953271607927442036502985349, 8.098395494836912586785942102816, 9.283327192509529222957087311172, 9.566680725194917737895040542757
