| L(s) = 1 | + 1.94·3-s − 2.68·5-s + 0.770·9-s − 0.616·11-s + 1.70·13-s − 5.21·15-s + 2.91·17-s − 1.49·19-s − 9.22·23-s + 2.21·25-s − 4.32·27-s + 10.4·29-s + 7.02·31-s − 1.19·33-s − 4.62·37-s + 3.30·39-s − 41-s + 4.88·43-s − 2.06·45-s + 4.96·47-s + 5.66·51-s − 8.14·53-s + 1.65·55-s − 2.90·57-s + 9.71·59-s + 7.04·61-s − 4.56·65-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 1.20·5-s + 0.256·9-s − 0.185·11-s + 0.471·13-s − 1.34·15-s + 0.707·17-s − 0.343·19-s − 1.92·23-s + 0.443·25-s − 0.833·27-s + 1.94·29-s + 1.26·31-s − 0.208·33-s − 0.760·37-s + 0.528·39-s − 0.156·41-s + 0.745·43-s − 0.308·45-s + 0.724·47-s + 0.792·51-s − 1.11·53-s + 0.223·55-s − 0.384·57-s + 1.26·59-s + 0.902·61-s − 0.566·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 + 2.68T + 5T^{2} \) |
| 11 | \( 1 + 0.616T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 + 9.22T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 - 4.96T + 47T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 59 | \( 1 - 9.71T + 59T^{2} \) |
| 61 | \( 1 - 7.04T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−7.893650784190140399254747129185, −6.95953694796545498407467810373, −6.20330572595799498444296464636, −5.36437981774304123043655880591, −4.17884562520955826651471673875, −4.02084767271942798378072017926, −3.05235699423200649684671778234, −2.51841585944807750910664689056, −1.31711524114578615084173497476, 0,
1.31711524114578615084173497476, 2.51841585944807750910664689056, 3.05235699423200649684671778234, 4.02084767271942798378072017926, 4.17884562520955826651471673875, 5.36437981774304123043655880591, 6.20330572595799498444296464636, 6.95953694796545498407467810373, 7.893650784190140399254747129185
