| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s − 11-s − 2·13-s + 16-s + 3·17-s − 2·19-s − 3·20-s − 22-s + 3·23-s + 4·25-s − 2·26-s − 2·31-s + 32-s + 3·34-s + 8·37-s − 2·38-s − 3·40-s + 9·41-s − 4·43-s − 44-s + 3·46-s − 3·47-s + 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s − 0.670·20-s − 0.213·22-s + 0.625·23-s + 4/5·25-s − 0.392·26-s − 0.359·31-s + 0.176·32-s + 0.514·34-s + 1.31·37-s − 0.324·38-s − 0.474·40-s + 1.40·41-s − 0.609·43-s − 0.150·44-s + 0.442·46-s − 0.437·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.37023479323627810325150062722, −6.74036650155956473279372101672, −5.85876212252901477737151827970, −5.18907036863904224366372528266, −4.40691442959840304003159553401, −3.96915662271737139631567843297, −3.10649126837413953544007017589, −2.50360154957090933781071266993, −1.20530789308196736052321812665, 0,
1.20530789308196736052321812665, 2.50360154957090933781071266993, 3.10649126837413953544007017589, 3.96915662271737139631567843297, 4.40691442959840304003159553401, 5.18907036863904224366372528266, 5.85876212252901477737151827970, 6.74036650155956473279372101672, 7.37023479323627810325150062722
