| L(s) = 1 | + 3-s + 9-s − 2·11-s + 2·13-s + 2·17-s + 4·19-s + 27-s + 2·29-s + 2·31-s − 2·33-s − 6·37-s + 2·39-s − 10·41-s − 8·43-s + 12·47-s − 7·49-s + 2·51-s + 8·53-s + 4·57-s + 10·59-s + 10·61-s + 8·67-s + 4·73-s + 10·79-s + 81-s − 4·83-s + 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.192·27-s + 0.371·29-s + 0.359·31-s − 0.348·33-s − 0.986·37-s + 0.320·39-s − 1.56·41-s − 1.21·43-s + 1.75·47-s − 49-s + 0.280·51-s + 1.09·53-s + 0.529·57-s + 1.30·59-s + 1.28·61-s + 0.977·67-s + 0.468·73-s + 1.12·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.714702069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.714702069\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.80368263810299683917762403250, −7.00662433002893786454181174893, −6.51440912683925255312701384289, −5.35833765571113529985517799135, −5.17337443675784408279497440993, −3.98204680449840151604539006084, −3.43337946589437071246516023518, −2.67743472317963424891594825907, −1.78206082006120931600636517156, −0.77784287161905984840430610839,
0.77784287161905984840430610839, 1.78206082006120931600636517156, 2.67743472317963424891594825907, 3.43337946589437071246516023518, 3.98204680449840151604539006084, 5.17337443675784408279497440993, 5.35833765571113529985517799135, 6.51440912683925255312701384289, 7.00662433002893786454181174893, 7.80368263810299683917762403250
