| L(s) = 1 | − 3-s − 3.56·5-s + 7-s + 9-s − 3.12·11-s + 13-s + 3.56·15-s + 1.12·17-s + 1.56·19-s − 21-s + 1.56·23-s + 7.68·25-s − 27-s − 7.56·29-s + 5.56·31-s + 3.12·33-s − 3.56·35-s + 1.12·37-s − 39-s − 2·41-s + 1.56·43-s − 3.56·45-s + 8.68·47-s + 49-s − 1.12·51-s − 1.31·53-s + 11.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.59·5-s + 0.377·7-s + 0.333·9-s − 0.941·11-s + 0.277·13-s + 0.919·15-s + 0.272·17-s + 0.358·19-s − 0.218·21-s + 0.325·23-s + 1.53·25-s − 0.192·27-s − 1.40·29-s + 0.998·31-s + 0.543·33-s − 0.602·35-s + 0.184·37-s − 0.160·39-s − 0.312·41-s + 0.238·43-s − 0.530·45-s + 1.26·47-s + 0.142·49-s − 0.157·51-s − 0.180·53-s + 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 1.56T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 0.876T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 0.438T + 73T^{2} \) |
| 79 | \( 1 + 0.684T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.87575271323689612241316906679, −7.44667140030201639559076286664, −6.68739668382502249755221281317, −5.62690127594722911619964936076, −5.03209765246110182136305336756, −4.20616277974553578246526970616, −3.56315743509505356394725820019, −2.54677968075550301463800410797, −1.08444517067964097977084642172, 0,
1.08444517067964097977084642172, 2.54677968075550301463800410797, 3.56315743509505356394725820019, 4.20616277974553578246526970616, 5.03209765246110182136305336756, 5.62690127594722911619964936076, 6.68739668382502249755221281317, 7.44667140030201639559076286664, 7.87575271323689612241316906679
