| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4948558091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4948558091\) |
| \(L(\frac{5}{2})\) |
|
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
−11.94684370035432418136501083309, −10.91502956215749865534688093002, −10.13172398623953892658652320473, −8.691893509727024553675133484698, −7.59082334020848234038393121575, −6.72581086983973173958572215135, −5.42389634698515555468196770023, −4.48524598379028397713416637779, −3.03833692115397508424811102457, −0.52704658593230053630683708530,
0.52704658593230053630683708530, 3.03833692115397508424811102457, 4.48524598379028397713416637779, 5.42389634698515555468196770023, 6.72581086983973173958572215135, 7.59082334020848234038393121575, 8.691893509727024553675133484698, 10.13172398623953892658652320473, 10.91502956215749865534688093002, 11.94684370035432418136501083309
