L(s) = 1 | + 1.73·3-s + 5i·5-s + 10.3·7-s + 2.99·9-s + 10.3i·11-s + 18i·13-s + 8.66i·15-s + 10i·17-s − 13.8i·19-s + 18·21-s − 6.92·23-s − 25·25-s + 5.19·27-s − 36·29-s + 6.92i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + i·5-s + 1.48·7-s + 0.333·9-s + 0.944i·11-s + 1.38i·13-s + 0.577i·15-s + 0.588i·17-s − 0.729i·19-s + 0.857·21-s − 0.301·23-s − 25-s + 0.192·27-s − 1.24·29-s + 0.223i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.615380562\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.615380562\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 - 5iT \) |
good | 7 | \( 1 - 10.3T + 49T^{2} \) |
| 11 | \( 1 - 10.3iT - 121T^{2} \) |
| 13 | \( 1 - 18iT - 169T^{2} \) |
| 17 | \( 1 - 10iT - 289T^{2} \) |
| 19 | \( 1 + 13.8iT - 361T^{2} \) |
| 23 | \( 1 + 6.92T + 529T^{2} \) |
| 29 | \( 1 + 36T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 54iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 26iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 41.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 36iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 90.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 90.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 18T + 7.92e3T^{2} \) |
| 97 | \( 1 + 72iT - 9.40e3T^{2} \) |
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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
−9.992338277248906737192648095612, −9.194374056996591074299170949409, −8.343458226474973262960203147887, −7.36412226168658085235076617001, −7.02296948193864081245242545617, −5.72957608371297792952821657852, −4.51550869231076721634850722952, −3.88193388633757651489564683164, −2.33010697967007716358992504207, −1.76423393508129699424519992716,
0.78196805635026846277212599965, 1.84753715509971178583210598063, 3.22160525526512845722656753553, 4.33392738028215068882207461963, 5.23686464897032305053045865667, 5.88177630943575908524729801212, 7.55737236823692193984324905845, 8.068937985410637363649364335998, 8.570535907640040429690992742639, 9.472345509248548992517221041577