| L(s) = 1 | − 0.523i·2-s + i·3-s + 1.72·4-s + 0.523·6-s + 3.20i·7-s − 1.95i·8-s − 9-s + 2.20·11-s + 1.72i·12-s + 3.04i·13-s + 1.67·14-s + 2.42·16-s + 0.523i·18-s − 19-s − 3.20·21-s − 1.15i·22-s + ⋯ |
| L(s) = 1 | − 0.370i·2-s + 0.577i·3-s + 0.862·4-s + 0.213·6-s + 1.21i·7-s − 0.690i·8-s − 0.333·9-s + 0.663·11-s + 0.498i·12-s + 0.845i·13-s + 0.448·14-s + 0.607·16-s + 0.123i·18-s − 0.229·19-s − 0.698·21-s − 0.245i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064059817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064059817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.523iT - 2T^{2} \) |
| 7 | \( 1 - 3.20iT - 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 - 3.04iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 23 | \( 1 - 3.24iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 3.45iT - 37T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 3.35iT - 47T^{2} \) |
| 53 | \( 1 - 0.904iT - 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 - 2.29iT - 67T^{2} \) |
| 71 | \( 1 - 7.29T + 71T^{2} \) |
| 73 | \( 1 - 1.09iT - 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 8.20iT - 83T^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 - 7.14iT - 97T^{2} \) |
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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
−9.502576837712285360049322334009, −9.224164814843266027291334522799, −8.197958488196315566786080528101, −7.18294604831151627345989608878, −6.26783308448763423894174435933, −5.68614273842355392757719915780, −4.50501872132703961025981465027, −3.51478549290231443997373577788, −2.56050305336771329513769182876, −1.60738563326235157121508974625,
0.833865326968442660366497955383, 2.05553751563134065071683262755, 3.22879120105861263629041757213, 4.24840154752848861300659770185, 5.52787503088387811870506868085, 6.29555378198488778435253578026, 7.06471444061845122381404946754, 7.56796198863814830952406323321, 8.296143607539727032434457488445, 9.330111492972477568199195718120
