| L(s) = 1 | − 1.84·2-s + 0.190·3-s + 1.40·4-s + 0.404·5-s − 0.352·6-s − 7-s + 1.09·8-s − 2.96·9-s − 0.747·10-s + 1.35·11-s + 0.268·12-s + 13-s + 1.84·14-s + 0.0772·15-s − 4.83·16-s − 17-s + 5.46·18-s + 5.97·19-s + 0.568·20-s − 0.190·21-s − 2.50·22-s − 1.48·23-s + 0.209·24-s − 4.83·25-s − 1.84·26-s − 1.13·27-s − 1.40·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.110·3-s + 0.702·4-s + 0.181·5-s − 0.143·6-s − 0.377·7-s + 0.388·8-s − 0.987·9-s − 0.236·10-s + 0.409·11-s + 0.0773·12-s + 0.277·13-s + 0.493·14-s + 0.0199·15-s − 1.20·16-s − 0.242·17-s + 1.28·18-s + 1.36·19-s + 0.127·20-s − 0.0416·21-s − 0.534·22-s − 0.310·23-s + 0.0427·24-s − 0.967·25-s − 0.361·26-s − 0.218·27-s − 0.265·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7276393653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7276393653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 3 | \( 1 - 0.190T + 3T^{2} \) |
| 5 | \( 1 - 0.404T + 5T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 + 1.48T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 + 6.53T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 + 7.60T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 - 2.18T + 67T^{2} \) |
| 71 | \( 1 + 3.38T + 71T^{2} \) |
| 73 | \( 1 + 4.33T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.192894225601649820798446332138, −8.998718453976729842778120014296, −7.86850293942570301419110978973, −7.45195766017243166299147504349, −6.29173029492820514733246944190, −5.60323735062276626460093169692, −4.32509788070823349646957159717, −3.18576505222552204071068295890, −2.04641532659661082029077685816, −0.72619481056459892713839576370,
0.72619481056459892713839576370, 2.04641532659661082029077685816, 3.18576505222552204071068295890, 4.32509788070823349646957159717, 5.60323735062276626460093169692, 6.29173029492820514733246944190, 7.45195766017243166299147504349, 7.86850293942570301419110978973, 8.998718453976729842778120014296, 9.192894225601649820798446332138
