| L(s) = 1 | − 3.46i·5-s + 4.89·7-s − 5.65i·11-s − 6.99·25-s + 10.3i·29-s − 4.89·31-s − 16.9i·35-s + 16.9·49-s + 3.46i·53-s − 19.5·55-s − 11.3i·59-s − 14·73-s − 27.7i·77-s + 14.6·79-s − 5.65i·83-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + 1.85·7-s − 1.70i·11-s − 1.39·25-s + 1.92i·29-s − 0.879·31-s − 2.86i·35-s + 2.42·49-s + 0.475i·53-s − 2.64·55-s − 1.47i·59-s − 1.63·73-s − 3.15i·77-s + 1.65·79-s − 0.620i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923417737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923417737\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 4.89T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.226839800686596881610336712553, −8.619073935047346454171987767225, −8.269930421563234859433188052655, −7.37574696715170511105871386254, −5.88577090978191828086898219279, −5.19400075369862367943554245804, −4.64128792556042520947209919546, −3.48434243962794387871118397399, −1.77515084512984858199377581218, −0.901531083708380012220976297629,
1.81426997628841063915823922505, 2.49880712103965964741820108707, 3.99260395541379466439485126254, 4.73751075772111442237944385832, 5.79672807323092104162910083789, 6.91129141325469602173581712488, 7.50959867217507741810557003114, 8.066359668354033505672001950517, 9.299931792174734302299512508018, 10.22445937317699142639592663251
