| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 2·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s + 17-s + 2·19-s + 2·20-s − 21-s − 23-s + 25-s − 27-s − 2·28-s + 8·29-s + 31-s − 2·33-s − 35-s − 2·36-s − 3·37-s + 5·39-s − 7·41-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.242·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + 0.179·31-s − 0.348·33-s − 0.169·35-s − 1/3·36-s − 0.493·37-s + 0.800·39-s − 1.09·41-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.875124152362556846665938617699, −8.130508438345186837574540048250, −7.36822758089724012978627819462, −6.50865430507889081738445221234, −5.38373779217897637905779872186, −4.79559864635169263531152538148, −4.11142735542990905129011155495, −2.98483251670407777110976435701, −1.33318755612277826546869106901, 0,
1.33318755612277826546869106901, 2.98483251670407777110976435701, 4.11142735542990905129011155495, 4.79559864635169263531152538148, 5.38373779217897637905779872186, 6.50865430507889081738445221234, 7.36822758089724012978627819462, 8.130508438345186837574540048250, 8.875124152362556846665938617699
