| L(s) = 1 | − i·3-s − 9-s − 4·11-s − 2i·13-s + 2i·17-s + 8·19-s + 4i·23-s + i·27-s − 6·29-s + 4i·33-s + 2i·37-s − 2·39-s − 6·41-s − 4i·43-s + 12i·47-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.333·9-s − 1.20·11-s − 0.554i·13-s + 0.485i·17-s + 1.83·19-s + 0.834i·23-s + 0.192i·27-s − 1.11·29-s + 0.696i·33-s + 0.328i·37-s − 0.320·39-s − 0.937·41-s − 0.609i·43-s + 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422015497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422015497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.046147729904683778177302242738, −7.64917687962721049091706370098, −7.13383000718153448427779313315, −6.00664242620333576473850816129, −5.50806447072765436742697522391, −4.85558668906089427779174144159, −3.56656064745155220738497737710, −2.97145665927498414875462796101, −1.95852100140149822520479181858, −0.897353534312629554796886580254,
0.47330227889760011102679101311, 1.97594621130449018447264489078, 2.91460347195458961443171633410, 3.63491282060755656464660176877, 4.62285788542302099364563090716, 5.28018375630313113262555174930, 5.79188931656388954270529950187, 6.97354995899325591917896807175, 7.43899866838358488122643349535, 8.298402926786489461057683659540
