| L(s) = 1 | + (−1 − i)2-s + 2i·4-s + (3 + 3i)7-s + (2 − 2i)8-s − 12·11-s + (12 − 12i)13-s − 6i·14-s − 4·16-s + (12 + 12i)17-s + 20i·19-s + (12 + 12i)22-s + (3 − 3i)23-s − 24·26-s + (−6 + 6i)28-s + 30i·29-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.5i)2-s + 0.5i·4-s + (0.428 + 0.428i)7-s + (0.250 − 0.250i)8-s − 1.09·11-s + (0.923 − 0.923i)13-s − 0.428i·14-s − 0.250·16-s + (0.705 + 0.705i)17-s + 1.05i·19-s + (0.545 + 0.545i)22-s + (0.130 − 0.130i)23-s − 0.923·26-s + (−0.214 + 0.214i)28-s + 1.03i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.307696399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307696399\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3 - 3i)T + 49iT^{2} \) |
| 11 | \( 1 + 12T + 121T^{2} \) |
| 13 | \( 1 + (-12 + 12i)T - 169iT^{2} \) |
| 17 | \( 1 + (-12 - 12i)T + 289iT^{2} \) |
| 19 | \( 1 - 20iT - 361T^{2} \) |
| 23 | \( 1 + (-3 + 3i)T - 529iT^{2} \) |
| 29 | \( 1 - 30iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 + (-48 - 48i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 48T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-27 + 27i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-27 - 27i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (12 - 12i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 60iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 48T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-12 + 12i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 40iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-93 + 93i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 30iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (12 + 12i)T + 9.40e3iT^{2} \) |
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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
−10.65070665597727047068646486991, −10.32077341497812238302166671179, −9.081880756409868057477351844910, −8.136274755116591802529441643843, −7.71580436606128620507935154608, −6.09959465441499361278566382897, −5.24063989221777173485652052904, −3.76251780683402980053370759450, −2.64197868092256633405935035009, −1.19029877160003910635911488839,
0.77792245781875828776542141108, 2.44953680416428328450775306204, 4.14065018367742846364065204204, 5.21717934895737897723979751904, 6.24793164507256641198846913758, 7.37623227060207952863675107331, 7.946680033049022121071910728037, 9.061516409740485338344520988386, 9.763698402816044382839187809700, 10.98816700885998946765037781109
