| L(s) = 1 | − 2i·7-s + 13.8·13-s + 27.7i·19-s − 25·25-s + 46i·31-s − 69.2·37-s − 83.1i·43-s + 45·49-s + 96.9·61-s + 55.4i·67-s − 46·73-s + 142i·79-s − 27.7i·91-s + 2·97-s + 194i·103-s + ⋯ |
| L(s) = 1 | − 0.285i·7-s + 1.06·13-s + 1.45i·19-s − 25-s + 1.48i·31-s − 1.87·37-s − 1.93i·43-s + 0.918·49-s + 1.59·61-s + 0.827i·67-s − 0.630·73-s + 1.79i·79-s − 0.304i·91-s + 0.0206·97-s + 1.88i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.475681705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475681705\) |
| \(L(2)\) |
|
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 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 2iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 13.8T + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 27.7iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 46iT - 961T^{2} \) |
| 37 | \( 1 + 69.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 83.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 96.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 55.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 142iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−8.746706825758170362346303802768, −8.453430895330631030155486698068, −7.43200620712702144022078586145, −6.74871299507659897025748801572, −5.82287087430057570229943984018, −5.21150718482105195636687613336, −3.87864323505332819700706137118, −3.56376982712620355006861496839, −2.09789687933273866820372978479, −1.13699750212752712994229667093,
0.38082544349490654396360220884, 1.69681759386098333015303043424, 2.75340168926415540908018792907, 3.73360162475811695293308427208, 4.59762212803831044866636257056, 5.56549297165530925564399277277, 6.25189007035115303042275961082, 7.07011194593175326177578218424, 7.927591381825613361685759523603, 8.683365907480909431181018394224
