| L(s) = 1 | − 3.37i·3-s + 2·4-s − 8.38·9-s − 6.74i·12-s + 4·16-s − 1.29i·17-s − 4.67i·19-s − 5.38·23-s − 5·25-s + 18.1i·27-s − 5.38·29-s − 5.44i·31-s − 16.7·36-s − 11.4i·41-s − 8.82i·47-s − 13.4i·48-s + ⋯ |
| L(s) = 1 | − 1.94i·3-s + 4-s − 2.79·9-s − 1.94i·12-s + 16-s − 0.315i·17-s − 1.07i·19-s − 1.12·23-s − 25-s + 3.49i·27-s − 1.00·29-s − 0.978i·31-s − 2.79·36-s − 1.78i·41-s − 1.28i·47-s − 1.94i·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534194631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534194631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 29 | \( 1 + 5.38T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + 3.37iT - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.29iT - 17T^{2} \) |
| 19 | \( 1 + 4.67iT - 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 31 | \( 1 + 5.44iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 11.4iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8.82iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14.7iT - 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 0.774iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 + 15.5iT - 97T^{2} \) |
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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
−8.842875466687023960583634590909, −8.021380972231543664864596421182, −7.35366284371582558770262995560, −6.91184901124406458959840678261, −6.01845934812948731081974007004, −5.49043782270745100912760161340, −3.62450109551829759489567693064, −2.37766907640515182174924106243, −1.96687228708635623723464391650, −0.55098905762593484278284305193,
2.03187306458562935893179764301, 3.28102467213452728119192838493, 3.82724595947184032610519009661, 4.87617425859207973323015263588, 5.81025758771523710409375243260, 6.32058028211758463037541295993, 7.83674858120868984608722296676, 8.313861207747474362838730107912, 9.581365400385088697420831557780, 9.855261840461110286935323477677
