| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 15.3·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 30.7·10-s + 29.6·11-s − 12·12-s + 9.57·13-s − 14·14-s + 46.1·15-s + 16·16-s − 102.·17-s − 18·18-s − 87.9·19-s − 61.4·20-s − 21·21-s − 59.3·22-s − 23·23-s + 24·24-s + 111.·25-s − 19.1·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.37·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.971·10-s + 0.812·11-s − 0.288·12-s + 0.204·13-s − 0.267·14-s + 0.793·15-s + 0.250·16-s − 1.45·17-s − 0.235·18-s − 1.06·19-s − 0.687·20-s − 0.218·21-s − 0.574·22-s − 0.208·23-s + 0.204·24-s + 0.889·25-s − 0.144·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 - 29.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.57T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 87.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 50.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 272.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 94.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 163.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 380.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 135.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 385.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 127.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.09e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.019364541776888691872288199685, −8.415461296788684747865762008838, −7.62474228597586650874133657749, −6.74921815844343281729134613543, −6.07559945152679969298609335006, −4.47146302604054809744940903072, −4.10260418364744233580913335169, −2.52189074790851742495117290898, −1.06252144540018010284169339464, 0,
1.06252144540018010284169339464, 2.52189074790851742495117290898, 4.10260418364744233580913335169, 4.47146302604054809744940903072, 6.07559945152679969298609335006, 6.74921815844343281729134613543, 7.62474228597586650874133657749, 8.415461296788684747865762008838, 9.019364541776888691872288199685
