| L(s) = 1 | − 4i·5-s + 11.3i·7-s − 14.1·11-s − 20i·13-s + 10·17-s + 14.1·19-s + 11.3i·23-s + 9·25-s + 20i·29-s + 45.2·35-s + 20i·37-s + 30·41-s + 2.82·43-s + 67.8i·47-s − 79.0·49-s + ⋯ |
| L(s) = 1 | − 0.800i·5-s + 1.61i·7-s − 1.28·11-s − 1.53i·13-s + 0.588·17-s + 0.744·19-s + 0.491i·23-s + 0.359·25-s + 0.689i·29-s + 1.29·35-s + 0.540i·37-s + 0.731·41-s + 0.0657·43-s + 1.44i·47-s − 1.61·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.613645053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613645053\) |
| \(L(2)\) |
|
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 | \( 1 + 4iT - 25T^{2} \) |
| 7 | \( 1 - 11.3iT - 49T^{2} \) |
| 11 | \( 1 + 14.1T + 121T^{2} \) |
| 13 | \( 1 + 20iT - 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 14.1T + 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 - 20iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 20iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.82T + 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 42.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 28iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 56.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 25.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 22T + 7.92e3T^{2} \) |
| 97 | \( 1 - 150T + 9.40e3T^{2} \) |
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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
−9.608706368424151494247997187819, −8.839018230205068744600902139479, −8.118745834822097100083701714058, −7.53814444851137930595741357977, −5.97740022532507760921388728871, −5.38912146803077385092951072245, −4.93971656391831153209905877939, −3.21308332616404734927944032533, −2.53629237100188405340807217166, −1.00596252537781928742871579000,
0.58978172035480969787101646321, 2.12434039780923255463240718749, 3.31682986872328799988832115238, 4.17806963697790233246542083425, 5.12460501401130447491245636206, 6.36140442746588439626312936191, 7.17852445754125627855625353360, 7.54912998651081804306397143027, 8.609998463014285403767602092591, 9.879527062259343889517054721462
