| L(s) = 1 | − 6i·3-s + 34i·7-s − 9·9-s − 16·11-s − 58i·13-s − 70i·17-s + 4·19-s + 204·21-s − 134i·23-s − 108i·27-s + 242·29-s − 100·31-s + 96i·33-s − 438i·37-s − 348·39-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 1.83i·7-s − 0.333·9-s − 0.438·11-s − 1.23i·13-s − 0.998i·17-s + 0.0482·19-s + 2.11·21-s − 1.21i·23-s − 0.769i·27-s + 1.54·29-s − 0.579·31-s + 0.506i·33-s − 1.94i·37-s − 1.42·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.473999443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473999443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 6iT - 27T^{2} \) |
| 7 | \( 1 - 34iT - 343T^{2} \) |
| 11 | \( 1 + 16T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 70iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 134iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 242T + 2.43e4T^{2} \) |
| 31 | \( 1 + 100T + 2.97e4T^{2} \) |
| 37 | \( 1 + 438iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 138T + 6.89e4T^{2} \) |
| 43 | \( 1 - 178iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 22iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 162iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 268T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 422iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 852T + 3.57e5T^{2} \) |
| 73 | \( 1 + 306iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 456T + 4.93e5T^{2} \) |
| 83 | \( 1 - 434iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 726T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3iT - 9.12e5T^{2} \) |
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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
−10.65826032989368437312712902917, −9.497745797918961760249217369403, −8.501489940770908291551762169589, −7.86478149262175088128876572911, −6.74282190826172659511437740234, −5.82456971045870520194256828519, −4.98514548673724351049126967314, −2.90170012563987265123658771305, −2.18412211548998227946560321303, −0.51723925781288068149057793760,
1.37921044625640459165007256554, 3.45748659073392774035359063936, 4.19442873377851438843916687330, 4.95670295008927289785875435569, 6.51582678129768632772835033799, 7.37929025720895743040115393577, 8.477831212631703126823508346588, 9.664062918747706105855953769166, 10.23063070210092321225062204636, 10.84647483471407206633882960493
