| L(s) = 1 | + 3.46i·5-s − 2·7-s − 1.73i·11-s − 4·13-s − 15.5i·17-s − 11·19-s + 27.7i·23-s + 13.0·25-s − 45.0i·29-s − 32·31-s − 6.92i·35-s − 34·37-s − 12.1i·41-s + 61·43-s − 48.4i·47-s + ⋯ |
| L(s) = 1 | + 0.692i·5-s − 0.285·7-s − 0.157i·11-s − 0.307·13-s − 0.916i·17-s − 0.578·19-s + 1.20i·23-s + 0.520·25-s − 1.55i·29-s − 1.03·31-s − 0.197i·35-s − 0.918·37-s − 0.295i·41-s + 1.41·43-s − 1.03i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.001878390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001878390\) |
| \(L(2)\) |
|
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 | \( 1 - 3.46iT - 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 1.73iT - 121T^{2} \) |
| 13 | \( 1 + 4T + 169T^{2} \) |
| 17 | \( 1 + 15.5iT - 289T^{2} \) |
| 19 | \( 1 + 11T + 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 45.0iT - 841T^{2} \) |
| 31 | \( 1 + 32T + 961T^{2} \) |
| 37 | \( 1 + 34T + 1.36e3T^{2} \) |
| 41 | \( 1 + 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 61T + 1.84e3T^{2} \) |
| 47 | \( 1 + 48.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56T + 3.72e3T^{2} \) |
| 67 | \( 1 - 31T + 4.48e3T^{2} \) |
| 71 | \( 1 + 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38T + 6.24e3T^{2} \) |
| 83 | \( 1 + 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 124. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115T + 9.40e3T^{2} \) |
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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
−9.411671771052165192746800276857, −8.461288725469633810835322432210, −7.45515078527246973222836518300, −6.91474511622387263210860611461, −5.96560014138101375008161867647, −5.10156127694762245778807808530, −3.93091865301527252882053056880, −3.03484075587231055403847347428, −2.02108164686237236106965675959, −0.30489943536045386666081858761,
1.17543726318088450328480598535, 2.41007357527634081730941328510, 3.65617619787777500507816703106, 4.59159809978453365461752437236, 5.39772286030385734806127657254, 6.40491751688745312791164680007, 7.15598319982150878316741748703, 8.240762066469506980399092864943, 8.815546951397439759505793585471, 9.546640761844739276704694706072
