| L(s) = 1 | − 2-s − 3-s + 4-s + 1.44·5-s + 6-s − 7-s − 8-s + 9-s − 1.44·10-s + 3.74·11-s − 12-s + 14-s − 1.44·15-s + 16-s + 0.692·17-s − 18-s + 0.753·19-s + 1.44·20-s + 21-s − 3.74·22-s + 1.82·23-s + 24-s − 2.91·25-s − 27-s − 28-s + 6.89·29-s + 1.44·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.646·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.456·10-s + 1.12·11-s − 0.288·12-s + 0.267·14-s − 0.373·15-s + 0.250·16-s + 0.167·17-s − 0.235·18-s + 0.172·19-s + 0.323·20-s + 0.218·21-s − 0.797·22-s + 0.381·23-s + 0.204·24-s − 0.582·25-s − 0.192·27-s − 0.188·28-s + 1.28·29-s + 0.263·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.405652396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405652396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 17 | \( 1 - 0.692T + 17T^{2} \) |
| 19 | \( 1 - 0.753T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 0.960T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + 1.44T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 + 1.57T + 53T^{2} \) |
| 59 | \( 1 + 6.65T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 + 0.219T + 71T^{2} \) |
| 73 | \( 1 - 4.91T + 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 6.06T + 97T^{2} \) |
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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
−7.900329585561927496966617030630, −7.19037690547333443514887775084, −6.41384287983191603786852875839, −6.14160838111282070179804425311, −5.28899718937192495046509167177, −4.38477249566000423743941385730, −3.50252655809650479940994392574, −2.52730519173745184241719212224, −1.55354456807299721256433767627, −0.73324093869823504627329628056,
0.73324093869823504627329628056, 1.55354456807299721256433767627, 2.52730519173745184241719212224, 3.50252655809650479940994392574, 4.38477249566000423743941385730, 5.28899718937192495046509167177, 6.14160838111282070179804425311, 6.41384287983191603786852875839, 7.19037690547333443514887775084, 7.900329585561927496966617030630
