| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 12.0·5-s + 6·6-s − 0.673·7-s + 8·8-s + 9·9-s − 24.1·10-s + 15.9·11-s + 12·12-s + 3.90·13-s − 1.34·14-s − 36.2·15-s + 16·16-s − 44.0·17-s + 18·18-s − 48.3·20-s − 2.02·21-s + 31.8·22-s − 9.11·23-s + 24·24-s + 21.1·25-s + 7.81·26-s + 27·27-s − 2.69·28-s − 161.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.08·5-s + 0.408·6-s − 0.0363·7-s + 0.353·8-s + 0.333·9-s − 0.764·10-s + 0.437·11-s + 0.288·12-s + 0.0834·13-s − 0.0257·14-s − 0.624·15-s + 0.250·16-s − 0.627·17-s + 0.235·18-s − 0.540·20-s − 0.0210·21-s + 0.309·22-s − 0.0826·23-s + 0.204·24-s + 0.169·25-s + 0.0589·26-s + 0.192·27-s − 0.0181·28-s − 1.03·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 12.0T + 125T^{2} \) |
| 7 | \( 1 + 0.673T + 343T^{2} \) |
| 11 | \( 1 - 15.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.90T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 9.11T + 1.21e4T^{2} \) |
| 29 | \( 1 + 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 60.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.90T + 7.95e4T^{2} \) |
| 47 | \( 1 - 145.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 754.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 32.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 77.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 44.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 280.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 779.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 274.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 832.T + 9.12e5T^{2} \) |
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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
−8.073453946281869208373139998544, −7.66205325662059052464281394390, −6.77492338296616773864077399924, −6.02506408762606047336468711322, −4.84176253361136468980567202819, −4.13694037839207776881827290322, −3.52262976430095927985305369391, −2.59916512531750744566313391627, −1.45363021171602645785431643037, 0,
1.45363021171602645785431643037, 2.59916512531750744566313391627, 3.52262976430095927985305369391, 4.13694037839207776881827290322, 4.84176253361136468980567202819, 6.02506408762606047336468711322, 6.77492338296616773864077399924, 7.66205325662059052464281394390, 8.073453946281869208373139998544
