| L(s) = 1 | − 2-s + 1.41i·3-s + 4-s − 1.41i·6-s − 2.64·7-s − 8-s + 0.999·9-s + 3.74i·11-s + 1.41i·12-s + 4.24i·13-s + 2.64·14-s + 16-s − 0.999·18-s − 5.29·19-s − 3.74i·21-s − 3.74i·22-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816i·3-s + 0.5·4-s − 0.577i·6-s − 0.999·7-s − 0.353·8-s + 0.333·9-s + 1.12i·11-s + 0.408i·12-s + 1.17i·13-s + 0.707·14-s + 0.250·16-s − 0.235·18-s − 1.21·19-s − 0.816i·21-s − 0.797i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.296758 + 0.618304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.296758 + 0.618304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + 2.64T \) |
| 23 | \( 1 + (3 + 3.74i)T \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.74iT - 53T^{2} \) |
| 59 | \( 1 + 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−11.96895982996578618770082863728, −10.51717120905856676271991472369, −10.11328691949484745623730741883, −9.336995673578719428254390439549, −8.478618341595021166014585116727, −7.00860497571241306211756392789, −6.46639347526639826809600464490, −4.75127959649727462015182895875, −3.78545433701755671865717783205, −2.10381351862526570217413915343,
0.59765340142038398772023628798, 2.40418026451990752365784642526, 3.76856629766051743070373772046, 5.87308308284396725950810075422, 6.42358553718720914610377248831, 7.65877560421752824588729103866, 8.246808524132898051611036485790, 9.484801200757308825072762352549, 10.24644666603516748781959407450, 11.21666698948593266522711476972
