| L(s) = 1 | − 1.61·3-s − 1.23·5-s + 4.85·7-s − 0.394·9-s − 1.78·11-s − 5.69·13-s + 1.99·15-s − 17-s + 2.51·19-s − 7.83·21-s + 6.91·23-s − 3.47·25-s + 5.47·27-s − 7.19·29-s + 5.87·31-s + 2.87·33-s − 5.98·35-s + 3.49·37-s + 9.19·39-s − 1.25·41-s + 1.45·43-s + 0.486·45-s + 2.43·47-s + 16.5·49-s + 1.61·51-s + 2.66·53-s + 2.19·55-s + ⋯ |
| L(s) = 1 | − 0.932·3-s − 0.552·5-s + 1.83·7-s − 0.131·9-s − 0.537·11-s − 1.57·13-s + 0.514·15-s − 0.242·17-s + 0.576·19-s − 1.70·21-s + 1.44·23-s − 0.695·25-s + 1.05·27-s − 1.33·29-s + 1.05·31-s + 0.500·33-s − 1.01·35-s + 0.574·37-s + 1.47·39-s − 0.195·41-s + 0.221·43-s + 0.0725·45-s + 0.354·47-s + 2.36·49-s + 0.226·51-s + 0.366·53-s + 0.296·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 + 7.19T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 - 3.49T + 37T^{2} \) |
| 41 | \( 1 + 1.25T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 + 9.97T + 73T^{2} \) |
| 79 | \( 1 - 1.01T + 79T^{2} \) |
| 83 | \( 1 + 2.01T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.60038914995642955828233394704, −6.97428462843155358734926137507, −5.85940844962796229312983790500, −5.26226668989182782100387067433, −4.80861716951887975877376126793, −4.29087142175975616833368605163, −2.98684290584353278522264295097, −2.18662360875008823934130395885, −1.09358943032154410070140605365, 0,
1.09358943032154410070140605365, 2.18662360875008823934130395885, 2.98684290584353278522264295097, 4.29087142175975616833368605163, 4.80861716951887975877376126793, 5.26226668989182782100387067433, 5.85940844962796229312983790500, 6.97428462843155358734926137507, 7.60038914995642955828233394704
