L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 4·13-s − 17-s + 5·19-s + 21-s − 9·23-s + 27-s + 9·29-s + 2·31-s − 33-s + 8·37-s + 4·39-s − 2·41-s + 11·43-s − 2·47-s + 49-s − 51-s + 3·53-s + 5·57-s − 59-s + 11·61-s + 63-s − 6·67-s − 9·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.242·17-s + 1.14·19-s + 0.218·21-s − 1.87·23-s + 0.192·27-s + 1.67·29-s + 0.359·31-s − 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 1.67·43-s − 0.291·47-s + 1/7·49-s − 0.140·51-s + 0.412·53-s + 0.662·57-s − 0.130·59-s + 1.40·61-s + 0.125·63-s − 0.733·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.837158257\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.837158257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.47523876374400, −14.08600980906108, −13.60145469772471, −13.33179424737237, −12.48335582064978, −12.11279863320192, −11.45226828023579, −11.08837811930855, −10.23649036253068, −10.04782861519681, −9.346934921694462, −8.729808310095162, −8.232089934573014, −7.877935674898815, −7.286645685744665, −6.542835833026173, −5.973723030361870, −5.505450899254935, −4.576352443481694, −4.203847717307245, −3.517157161054717, −2.802243255735576, −2.243877461831765, −1.370852316712585, −0.7217275580251307,
0.7217275580251307, 1.370852316712585, 2.243877461831765, 2.802243255735576, 3.517157161054717, 4.203847717307245, 4.576352443481694, 5.505450899254935, 5.973723030361870, 6.542835833026173, 7.286645685744665, 7.877935674898815, 8.232089934573014, 8.729808310095162, 9.346934921694462, 10.04782861519681, 10.23649036253068, 11.08837811930855, 11.45226828023579, 12.11279863320192, 12.48335582064978, 13.33179424737237, 13.60145469772471, 14.08600980906108, 14.47523876374400