L(s) = 1 | + 2·2-s + 2·4-s + 3·7-s + 5·11-s − 13-s + 6·14-s − 4·16-s + 5·17-s + 2·19-s + 10·22-s − 23-s − 2·26-s + 6·28-s − 10·29-s − 2·31-s − 8·32-s + 10·34-s + 3·37-s + 4·38-s + 9·41-s + 4·43-s + 10·44-s − 2·46-s + 10·47-s + 2·49-s − 2·52-s + 9·53-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 1.13·7-s + 1.50·11-s − 0.277·13-s + 1.60·14-s − 16-s + 1.21·17-s + 0.458·19-s + 2.13·22-s − 0.208·23-s − 0.392·26-s + 1.13·28-s − 1.85·29-s − 0.359·31-s − 1.41·32-s + 1.71·34-s + 0.493·37-s + 0.648·38-s + 1.40·41-s + 0.609·43-s + 1.50·44-s − 0.294·46-s + 1.45·47-s + 2/7·49-s − 0.277·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.848100773\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.848100773\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−8.927414006080760042443813551892, −7.60803187099497637204674787562, −7.31077845810985799511898034435, −6.01156418618439851476188424220, −5.71044508613950962582279010712, −4.76718252747174206383016384784, −4.08049179866591711849168626239, −3.46601352996457060594826979316, −2.28197649627513544566528453151, −1.22859988534775401619018553396,
1.22859988534775401619018553396, 2.28197649627513544566528453151, 3.46601352996457060594826979316, 4.08049179866591711849168626239, 4.76718252747174206383016384784, 5.71044508613950962582279010712, 6.01156418618439851476188424220, 7.31077845810985799511898034435, 7.60803187099497637204674787562, 8.927414006080760042443813551892