| L(s) = 1 | − 2.05·2-s − 3.75·4-s − 14.5·5-s + 24.2·8-s + 29.8·10-s − 29.8·11-s − 13.3·13-s − 19.7·16-s + 64.2·17-s − 110.·19-s + 54.5·20-s + 61.3·22-s − 19.3·23-s + 85.3·25-s + 27.4·26-s + 111.·29-s + 192.·31-s − 152.·32-s − 132.·34-s − 71.5·37-s + 227.·38-s − 351.·40-s + 277.·41-s − 178.·43-s + 112.·44-s + 39.8·46-s + 531.·47-s + ⋯ |
| L(s) = 1 | − 0.728·2-s − 0.469·4-s − 1.29·5-s + 1.07·8-s + 0.944·10-s − 0.817·11-s − 0.284·13-s − 0.309·16-s + 0.917·17-s − 1.33·19-s + 0.609·20-s + 0.594·22-s − 0.175·23-s + 0.682·25-s + 0.207·26-s + 0.711·29-s + 1.11·31-s − 0.844·32-s − 0.667·34-s − 0.317·37-s + 0.970·38-s − 1.38·40-s + 1.05·41-s − 0.633·43-s + 0.383·44-s + 0.127·46-s + 1.65·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.05T + 8T^{2} \) |
| 5 | \( 1 + 14.5T + 125T^{2} \) |
| 11 | \( 1 + 29.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 277.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 178.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 722.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 663.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 608.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 976.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 791.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 935.T + 9.12e5T^{2} \) |
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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
−8.682849696145660827140494961354, −8.004207726995912784100311856622, −7.69071843024610099870415836489, −6.63161931346524311809211519907, −5.33368599831190043425442093392, −4.46231177138341515217118411778, −3.76514500587472660682676713347, −2.48602901327919660557542862417, −0.900100721732320072677139453255, 0,
0.900100721732320072677139453255, 2.48602901327919660557542862417, 3.76514500587472660682676713347, 4.46231177138341515217118411778, 5.33368599831190043425442093392, 6.63161931346524311809211519907, 7.69071843024610099870415836489, 8.004207726995912784100311856622, 8.682849696145660827140494961354
