| L(s) = 1 | + 1.11·2-s + 0.728·3-s − 6.76·4-s + 6.16·5-s + 0.810·6-s − 4.11·7-s − 16.4·8-s − 26.4·9-s + 6.86·10-s − 4.92·12-s − 13·13-s − 4.57·14-s + 4.49·15-s + 35.8·16-s + 97.9·17-s − 29.4·18-s − 62.5·19-s − 41.7·20-s − 3.00·21-s − 34.1·23-s − 11.9·24-s − 86.9·25-s − 14.4·26-s − 38.9·27-s + 27.8·28-s − 194.·29-s + 5.00·30-s + ⋯ |
| L(s) = 1 | + 0.393·2-s + 0.140·3-s − 0.845·4-s + 0.551·5-s + 0.0551·6-s − 0.222·7-s − 0.725·8-s − 0.980·9-s + 0.216·10-s − 0.118·12-s − 0.277·13-s − 0.0873·14-s + 0.0773·15-s + 0.560·16-s + 1.39·17-s − 0.385·18-s − 0.755·19-s − 0.466·20-s − 0.0311·21-s − 0.309·23-s − 0.101·24-s − 0.695·25-s − 0.109·26-s − 0.277·27-s + 0.187·28-s − 1.24·29-s + 0.0304·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.562066631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.562066631\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 1.11T + 8T^{2} \) |
| 3 | \( 1 - 0.728T + 27T^{2} \) |
| 5 | \( 1 - 6.16T + 125T^{2} \) |
| 7 | \( 1 + 4.11T + 343T^{2} \) |
| 17 | \( 1 - 97.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 62.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 36.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 16.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 139.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 81.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 542.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 288.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−9.171097051899893150403260824994, −8.300595649726647155030224655841, −7.65641366895200931903355552343, −6.29293393793783415585881163924, −5.71519734765721401459454238478, −5.05667129657340380414830003774, −3.91524698043747258566689371510, −3.17611992087063884864121265867, −2.07230759516998590002590288017, −0.55143720598340768187064952670,
0.55143720598340768187064952670, 2.07230759516998590002590288017, 3.17611992087063884864121265867, 3.91524698043747258566689371510, 5.05667129657340380414830003774, 5.71519734765721401459454238478, 6.29293393793783415585881163924, 7.65641366895200931903355552343, 8.300595649726647155030224655841, 9.171097051899893150403260824994
