| L(s) = 1 | + 2·5-s + 7-s + 2·11-s − 13-s − 6·17-s + 5·19-s + 6·23-s − 25-s + 8·29-s + 8·31-s + 2·35-s + 5·37-s − 8·41-s + 4·43-s − 10·47-s − 6·49-s + 4·53-s + 4·55-s − 14·59-s − 3·61-s − 2·65-s + 13·67-s − 4·71-s + 9·73-s + 2·77-s + 11·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 0.603·11-s − 0.277·13-s − 1.45·17-s + 1.14·19-s + 1.25·23-s − 1/5·25-s + 1.48·29-s + 1.43·31-s + 0.338·35-s + 0.821·37-s − 1.24·41-s + 0.609·43-s − 1.45·47-s − 6/7·49-s + 0.549·53-s + 0.539·55-s − 1.82·59-s − 0.384·61-s − 0.248·65-s + 1.58·67-s − 0.474·71-s + 1.05·73-s + 0.227·77-s + 1.23·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.244027973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244027973\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.406695899512768329172165700211, −8.628160865153058100643452862475, −7.79687737318869765033439112225, −6.66685854892167006683283143935, −6.33133264445040476268419802877, −5.07185708630422123894026254024, −4.60692733212276996243667777217, −3.22264556178247491549150269163, −2.24206477413057765543268643363, −1.11475621362688095791673953931,
1.11475621362688095791673953931, 2.24206477413057765543268643363, 3.22264556178247491549150269163, 4.60692733212276996243667777217, 5.07185708630422123894026254024, 6.33133264445040476268419802877, 6.66685854892167006683283143935, 7.79687737318869765033439112225, 8.628160865153058100643452862475, 9.406695899512768329172165700211
