| L(s) = 1 | + 3-s + 7-s + 9-s − 2·13-s + 6·17-s + 4·19-s + 21-s + 27-s − 6·29-s + 4·31-s − 2·37-s − 2·39-s + 6·41-s + 8·43-s − 12·47-s + 49-s + 6·51-s − 6·53-s + 4·57-s + 12·59-s + 2·61-s + 63-s + 8·67-s − 14·73-s + 16·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-s + 0.977·67-s − 1.63·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.940836908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.940836908\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.81871125561743184250629958049, −7.33189550653539899010272884155, −6.50762570864184055543638072634, −5.54704673433490752148991068707, −5.11024302312920159334311275843, −4.16611950153165149285581206648, −3.42598458660049442584897651736, −2.72315856559003551216690246233, −1.78774388725724014340172002076, −0.847218896403025901538886718011,
0.847218896403025901538886718011, 1.78774388725724014340172002076, 2.72315856559003551216690246233, 3.42598458660049442584897651736, 4.16611950153165149285581206648, 5.11024302312920159334311275843, 5.54704673433490752148991068707, 6.50762570864184055543638072634, 7.33189550653539899010272884155, 7.81871125561743184250629958049
