| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 20.1·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 40.3·10-s − 63.4·11-s − 12·12-s − 30.9·13-s − 14·14-s + 60.4·15-s + 16·16-s + 84.3·17-s − 18·18-s + 147.·19-s − 80.6·20-s − 21·21-s + 126.·22-s − 23·23-s + 24·24-s + 281.·25-s + 61.8·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.80·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.27·10-s − 1.74·11-s − 0.288·12-s − 0.659·13-s − 0.267·14-s + 1.04·15-s + 0.250·16-s + 1.20·17-s − 0.235·18-s + 1.78·19-s − 0.901·20-s − 0.218·21-s + 1.23·22-s − 0.208·23-s + 0.204·24-s + 2.25·25-s + 0.466·26-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 | \( 1 + 20.1T + 125T^{2} \) |
| 11 | \( 1 + 63.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 88.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 370.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 414.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 12.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 579.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 973.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 397.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 29.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 398.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 981.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 870.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 19.8T + 9.12e5T^{2} \) |
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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
−9.235294543799263157322594836273, −7.938080438941655822768624962598, −7.70664418643667496670820239823, −7.21109548225472067373150315193, −5.58009227602925209543886125957, −4.97265486912443302374699517779, −3.71876960220437373363000937293, −2.71968982496573781161177519726, −0.932177518834570060546126725428, 0,
0.932177518834570060546126725428, 2.71968982496573781161177519726, 3.71876960220437373363000937293, 4.97265486912443302374699517779, 5.58009227602925209543886125957, 7.21109548225472067373150315193, 7.70664418643667496670820239823, 7.938080438941655822768624962598, 9.235294543799263157322594836273
