L(s) = 1 | + 0.847·2-s + 3-s − 1.28·4-s − 4.35·5-s + 0.847·6-s − 7-s − 2.78·8-s + 9-s − 3.68·10-s + 0.958·11-s − 1.28·12-s + 1.94·13-s − 0.847·14-s − 4.35·15-s + 0.206·16-s + 4.26·17-s + 0.847·18-s − 3.64·19-s + 5.57·20-s − 21-s + 0.812·22-s − 3.28·23-s − 2.78·24-s + 13.9·25-s + 1.65·26-s + 27-s + 1.28·28-s + ⋯ |
L(s) = 1 | + 0.599·2-s + 0.577·3-s − 0.640·4-s − 1.94·5-s + 0.346·6-s − 0.377·7-s − 0.983·8-s + 0.333·9-s − 1.16·10-s + 0.288·11-s − 0.369·12-s + 0.540·13-s − 0.226·14-s − 1.12·15-s + 0.0515·16-s + 1.03·17-s + 0.199·18-s − 0.835·19-s + 1.24·20-s − 0.218·21-s + 0.173·22-s − 0.684·23-s − 0.567·24-s + 2.78·25-s + 0.323·26-s + 0.192·27-s + 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 0.847T + 2T^{2} \) |
| 5 | \( 1 + 4.35T + 5T^{2} \) |
| 11 | \( 1 - 0.958T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 3.64T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 + 0.0219T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 + 0.00377T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 7.86T + 61T^{2} \) |
| 67 | \( 1 - 2.22T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 9.59T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 9.51T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.62945178489917430835725983591, −6.87473373621940283575323353202, −6.07948150975611378811497741050, −5.18307369292554404771795471048, −4.31075408016720288069817628051, −3.89879093764879117657299908533, −3.45654006484442598117337668048, −2.67480803769868515065530539376, −1.04769881376796038192290337270, 0,
1.04769881376796038192290337270, 2.67480803769868515065530539376, 3.45654006484442598117337668048, 3.89879093764879117657299908533, 4.31075408016720288069817628051, 5.18307369292554404771795471048, 6.07948150975611378811497741050, 6.87473373621940283575323353202, 7.62945178489917430835725983591