| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + 2.82i·5-s − 2.82i·8-s + (2.00 − 3.46i)10-s − 5.65i·11-s + 5.19i·13-s + (−2.00 + 3.46i)16-s + 2.82i·17-s + 5·19-s + (−4.89 + 2.82i)20-s + (−4.00 + 6.92i)22-s + 2.82i·23-s − 3.00·25-s + (3.67 − 6.36i)26-s + ⋯ |
| L(s) = 1 | + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + 1.26i·5-s − 0.999i·8-s + (0.632 − 1.09i)10-s − 1.70i·11-s + 1.44i·13-s + (−0.500 + 0.866i)16-s + 0.685i·17-s + 1.14·19-s + (−1.09 + 0.632i)20-s + (−0.852 + 1.47i)22-s + 0.589i·23-s − 0.600·25-s + (0.720 − 1.24i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9726719180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9726719180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.444892262873631910073828592794, −8.833496543320452512136660544838, −7.944197220377311877898379598859, −7.19906401600566278905645563188, −6.48030074634034173275525992052, −5.74507713968766443827240559416, −4.05679103195680753470309289927, −3.32004293719380969234598330865, −2.54988680114651465770006326286, −1.25769634733280625088146229130,
0.54132136736487964541055395384, 1.61071536390437521291227571975, 2.85907663482740922216434955025, 4.56751673421086963884714765289, 5.08491924441494207484496255518, 5.86039875585906350672168207392, 7.03292499968141347750761143713, 7.66982798716928210953138224039, 8.268848752646218983229521013053, 9.211937502999426367356716833919
