L(s) = 1 | − 5.90·3-s + 14.9·5-s + 6.82·7-s + 7.81·9-s − 18.6·11-s + 39.8·13-s − 88.4·15-s + 19.3·17-s + 2.14·19-s − 40.2·21-s − 103.·23-s + 99.9·25-s + 113.·27-s − 29·29-s − 213.·31-s + 110.·33-s + 102.·35-s + 226.·37-s − 235.·39-s − 191.·41-s + 34.3·43-s + 117.·45-s − 501.·47-s − 296.·49-s − 114.·51-s − 259.·53-s − 280.·55-s + ⋯ |
L(s) = 1 | − 1.13·3-s + 1.34·5-s + 0.368·7-s + 0.289·9-s − 0.512·11-s + 0.850·13-s − 1.52·15-s + 0.275·17-s + 0.0259·19-s − 0.418·21-s − 0.942·23-s + 0.799·25-s + 0.807·27-s − 0.185·29-s − 1.23·31-s + 0.581·33-s + 0.494·35-s + 1.00·37-s − 0.965·39-s − 0.728·41-s + 0.121·43-s + 0.388·45-s − 1.55·47-s − 0.864·49-s − 0.313·51-s − 0.673·53-s − 0.687·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
good | 3 | \( 1 + 5.90T + 27T^{2} \) |
| 5 | \( 1 - 14.9T + 125T^{2} \) |
| 7 | \( 1 - 6.82T + 343T^{2} \) |
| 11 | \( 1 + 18.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.14T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 226.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 191.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 34.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 501.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 259.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 81.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 799.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 575.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 22.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 118.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 235.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 295.T + 9.12e5T^{2} \) |
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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
−8.515224784754819850686776499308, −7.68159371306768354911635666347, −6.49844310181043921275596806976, −6.03230199071975524981079756245, −5.41102050251651284424776949964, −4.73324133219744532869321327661, −3.41321538541475408996948488114, −2.13856921259137118218701907335, −1.29143581509575557557306871291, 0,
1.29143581509575557557306871291, 2.13856921259137118218701907335, 3.41321538541475408996948488114, 4.73324133219744532869321327661, 5.41102050251651284424776949964, 6.03230199071975524981079756245, 6.49844310181043921275596806976, 7.68159371306768354911635666347, 8.515224784754819850686776499308