| L(s) = 1 | + 3.31·3-s + 7.99·9-s − 4.24·11-s − 5.10·13-s + 0.883·17-s − 6.54·19-s − 5.33·23-s + 16.5·27-s + 4.54·29-s + 0.677·31-s − 14.0·33-s − 1.52·37-s − 16.9·39-s − 5.92·41-s + 5.49·43-s − 7.26·47-s + 2.92·51-s − 2.56·53-s − 21.6·57-s − 7.94·59-s − 9.01·61-s − 3.04·67-s − 17.6·69-s + 5.39·71-s − 3.85·73-s + 0.704·79-s + 30.9·81-s + ⋯ |
| L(s) = 1 | + 1.91·3-s + 2.66·9-s − 1.28·11-s − 1.41·13-s + 0.214·17-s − 1.50·19-s − 1.11·23-s + 3.18·27-s + 0.843·29-s + 0.121·31-s − 2.45·33-s − 0.250·37-s − 2.71·39-s − 0.925·41-s + 0.838·43-s − 1.06·47-s + 0.410·51-s − 0.352·53-s − 2.87·57-s − 1.03·59-s − 1.15·61-s − 0.371·67-s − 2.12·69-s + 0.640·71-s − 0.451·73-s + 0.0792·79-s + 3.43·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 0.883T + 17T^{2} \) |
| 19 | \( 1 + 6.54T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 - 0.677T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 + 5.92T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 + 7.26T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 + 9.01T + 61T^{2} \) |
| 67 | \( 1 + 3.04T + 67T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 + 3.85T + 73T^{2} \) |
| 79 | \( 1 - 0.704T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 8.40T + 89T^{2} \) |
| 97 | \( 1 + 0.614T + 97T^{2} \) |
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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
−7.58830175213622559821185645975, −6.93735363967461769926772751656, −6.11608747423994407419442910759, −4.90057305165983246131310722800, −4.53100469546927015169520071394, −3.64007059141125539874653707983, −2.82220375070645217073763709453, −2.37702974638330469847983146718, −1.69023455999897669829204606900, 0,
1.69023455999897669829204606900, 2.37702974638330469847983146718, 2.82220375070645217073763709453, 3.64007059141125539874653707983, 4.53100469546927015169520071394, 4.90057305165983246131310722800, 6.11608747423994407419442910759, 6.93735363967461769926772751656, 7.58830175213622559821185645975
