| L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 2·11-s − 7·13-s + 4·17-s − 6·19-s + 4·21-s + 23-s − 5·27-s + 5·29-s + 3·31-s − 2·33-s − 2·37-s − 7·39-s − 9·41-s − 8·43-s + 47-s + 9·49-s + 4·51-s + 6·53-s − 6·57-s − 8·59-s − 10·61-s − 8·63-s − 2·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.603·11-s − 1.94·13-s + 0.970·17-s − 1.37·19-s + 0.872·21-s + 0.208·23-s − 0.962·27-s + 0.928·29-s + 0.538·31-s − 0.348·33-s − 0.328·37-s − 1.12·39-s − 1.40·41-s − 1.21·43-s + 0.145·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.794·57-s − 1.04·59-s − 1.28·61-s − 1.00·63-s − 0.244·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.034387861490633451898627491741, −7.51575267109449801241987108949, −6.62872930281327627027285953667, −5.50610552003431010359051451218, −4.96636573458812700033501711884, −4.38605040643404791188637200745, −3.10516585216650234415273937973, −2.44634667518859626375593246595, −1.64047907913666212564911496252, 0,
1.64047907913666212564911496252, 2.44634667518859626375593246595, 3.10516585216650234415273937973, 4.38605040643404791188637200745, 4.96636573458812700033501711884, 5.50610552003431010359051451218, 6.62872930281327627027285953667, 7.51575267109449801241987108949, 8.034387861490633451898627491741
