| L(s) = 1 | − 3-s + 1.46·5-s + 2.34·7-s + 9-s − 2.63·11-s + 2·13-s − 1.46·15-s + 1.61·17-s − 3.41·19-s − 2.34·21-s − 8·23-s − 2.86·25-s − 27-s − 7.26·29-s + 3.41·31-s + 2.63·33-s + 3.41·35-s + 0.340·37-s − 2·39-s − 8.63·41-s − 7.95·43-s + 1.46·45-s − 3.89·47-s − 1.52·49-s − 1.61·51-s + 6.29·53-s − 3.84·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.653·5-s + 0.884·7-s + 0.333·9-s − 0.793·11-s + 0.554·13-s − 0.377·15-s + 0.392·17-s − 0.784·19-s − 0.510·21-s − 1.66·23-s − 0.573·25-s − 0.192·27-s − 1.34·29-s + 0.613·31-s + 0.457·33-s + 0.577·35-s + 0.0559·37-s − 0.320·39-s − 1.34·41-s − 1.21·43-s + 0.217·45-s − 0.567·47-s − 0.217·49-s − 0.226·51-s + 0.865·53-s − 0.518·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.46T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 - 0.340T + 37T^{2} \) |
| 41 | \( 1 + 8.63T + 41T^{2} \) |
| 43 | \( 1 + 7.95T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 - 6.29T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 - 2.35T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−8.117556756463573049450225867499, −7.41506980895942642208548855041, −6.40866250910239576938667840920, −5.81381608265759520226319229916, −5.22014281052842538230780686933, −4.40943342200377433208895373556, −3.51203875389296938405926345205, −2.15652732782009961347848103472, −1.58494208720156312370006693725, 0,
1.58494208720156312370006693725, 2.15652732782009961347848103472, 3.51203875389296938405926345205, 4.40943342200377433208895373556, 5.22014281052842538230780686933, 5.81381608265759520226319229916, 6.40866250910239576938667840920, 7.41506980895942642208548855041, 8.117556756463573049450225867499
