| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 17.9·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s + 35.8·10-s + 37.7·11-s + 12·12-s − 43.6·13-s + 14·14-s − 53.7·15-s + 16·16-s + 94.6·17-s − 18·18-s − 2.78·19-s − 71.6·20-s − 21·21-s − 75.4·22-s − 23·23-s − 24·24-s + 195.·25-s + 87.2·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.60·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.13·10-s + 1.03·11-s + 0.288·12-s − 0.930·13-s + 0.267·14-s − 0.924·15-s + 0.250·16-s + 1.35·17-s − 0.235·18-s − 0.0336·19-s − 0.800·20-s − 0.218·21-s − 0.731·22-s − 0.208·23-s − 0.204·24-s + 1.56·25-s + 0.657·26-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 | \( 1 + 17.9T + 125T^{2} \) |
| 11 | \( 1 - 37.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.78T + 6.85e3T^{2} \) |
| 29 | \( 1 + 84.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 57.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 94.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 547.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 440.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 176.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 572.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 157.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 452.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 950.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 946.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 920.T + 9.12e5T^{2} \) |
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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
−9.156728764835982263969742243054, −8.360329794328114516760179628346, −7.50346785798058726818174487411, −7.22830744330518769532439727660, −5.97151636275458924360095425325, −4.48119530247518451967125360836, −3.65645952573521579604031897377, −2.80507694017243662967691076203, −1.22266543281403195085415495765, 0,
1.22266543281403195085415495765, 2.80507694017243662967691076203, 3.65645952573521579604031897377, 4.48119530247518451967125360836, 5.97151636275458924360095425325, 7.22830744330518769532439727660, 7.50346785798058726818174487411, 8.360329794328114516760179628346, 9.156728764835982263969742243054
