| L(s) = 1 | + 7.37i·5-s − 1.26·7-s − 5.82i·11-s + 10.4·13-s − 18.3i·17-s − 20.8·19-s − 20.4i·23-s − 29.4·25-s + 11.1i·29-s − 61.3·31-s − 9.34i·35-s + 38.4·37-s + 33.0i·41-s − 49.3·43-s + 21.5i·47-s + ⋯ |
| L(s) = 1 | + 1.47i·5-s − 0.180·7-s − 0.529i·11-s + 0.806·13-s − 1.07i·17-s − 1.09·19-s − 0.890i·23-s − 1.17·25-s + 0.385i·29-s − 1.97·31-s − 0.266i·35-s + 1.03·37-s + 0.807i·41-s − 1.14·43-s + 0.459i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8340102984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8340102984\) |
| \(L(2)\) |
|
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 | \( 1 - 7.37iT - 25T^{2} \) |
| 7 | \( 1 + 1.26T + 49T^{2} \) |
| 11 | \( 1 + 5.82iT - 121T^{2} \) |
| 13 | \( 1 - 10.4T + 169T^{2} \) |
| 17 | \( 1 + 18.3iT - 289T^{2} \) |
| 19 | \( 1 + 20.8T + 361T^{2} \) |
| 23 | \( 1 + 20.4iT - 529T^{2} \) |
| 29 | \( 1 - 11.1iT - 841T^{2} \) |
| 31 | \( 1 + 61.3T + 961T^{2} \) |
| 37 | \( 1 - 38.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 55.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 91.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 114. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 8.21T + 6.24e3T^{2} \) |
| 83 | \( 1 + 150. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.8T + 9.40e3T^{2} \) |
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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
−8.924579284881180654854949634798, −8.128914739196072633158104183288, −7.19168298323002839016428297539, −6.54284619305867718926143376643, −5.97947287119936085330792297712, −4.79342034205748495532854036515, −3.60908740425306774205478729141, −3.01267039141437514057938849058, −1.96188380626474678422537689022, −0.22444004685550114360863788622,
1.20199136471032813362627915931, 2.05513623801203631181633043330, 3.72426595086213498146653673317, 4.26793420728537021747199535403, 5.32199512066190494757286485540, 5.93405729174165966284401924375, 6.95010088620218261113371955932, 7.996847153197974733642650327436, 8.574919372979746360458182323768, 9.203530033780612271725956222370
