| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·11-s − 5·13-s − 14-s + 16-s − 7·19-s − 3·22-s + 6·23-s − 5·26-s − 28-s + 6·29-s − 4·31-s + 32-s − 2·37-s − 7·38-s + 3·41-s + 43-s − 3·44-s + 6·46-s + 3·47-s + 49-s − 5·52-s + 9·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.904·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 1.60·19-s − 0.639·22-s + 1.25·23-s − 0.980·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.328·37-s − 1.13·38-s + 0.468·41-s + 0.152·43-s − 0.452·44-s + 0.884·46-s + 0.437·47-s + 1/7·49-s − 0.693·52-s + 1.23·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.283866516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283866516\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.49151084043621296972244930273, −6.95741491601282454989064003122, −6.36212916146949208415830759300, −5.47640836619056107584606497782, −4.95175046706741470199807242344, −4.32794457128924352531589771702, −3.43220546399247587766441208690, −2.56614055867355040241680025605, −2.16179865689275459250940983139, −0.61150503212047106535072987527,
0.61150503212047106535072987527, 2.16179865689275459250940983139, 2.56614055867355040241680025605, 3.43220546399247587766441208690, 4.32794457128924352531589771702, 4.95175046706741470199807242344, 5.47640836619056107584606497782, 6.36212916146949208415830759300, 6.95741491601282454989064003122, 7.49151084043621296972244930273
