| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 8.44·5-s − 6·6-s − 9.44·7-s + 8·8-s + 9·9-s − 16.8·10-s − 47.4·11-s − 12·12-s + 67.6·13-s − 18.8·14-s + 25.3·15-s + 16·16-s − 77.6·17-s + 18·18-s − 33.7·20-s + 28.3·21-s − 94.9·22-s + 170.·23-s − 24·24-s − 53.7·25-s + 135.·26-s − 27·27-s − 37.7·28-s + 240.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.755·5-s − 0.408·6-s − 0.509·7-s + 0.353·8-s + 0.333·9-s − 0.533·10-s − 1.30·11-s − 0.288·12-s + 1.44·13-s − 0.360·14-s + 0.435·15-s + 0.250·16-s − 1.10·17-s + 0.235·18-s − 0.377·20-s + 0.294·21-s − 0.920·22-s + 1.54·23-s − 0.204·24-s − 0.429·25-s + 1.02·26-s − 0.192·27-s − 0.254·28-s + 1.53·29-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 | \( 1 + 8.44T + 125T^{2} \) |
| 7 | \( 1 + 9.44T + 343T^{2} \) |
| 11 | \( 1 + 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.6T + 4.91e3T^{2} \) |
| 23 | \( 1 - 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 71.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 207.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 261.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 21.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 413.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 157.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 417.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 138.T + 9.12e5T^{2} \) |
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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
−8.204682954127224865447567993561, −7.44905230758568324751638489480, −6.47134108718901015267226278878, −6.11535773269686048163206627505, −4.92060896877526683111757824735, −4.46731649630910423975293776314, −3.37147577894433230143714113089, −2.67005346838084301496189808136, −1.14651442190544055386270054683, 0,
1.14651442190544055386270054683, 2.67005346838084301496189808136, 3.37147577894433230143714113089, 4.46731649630910423975293776314, 4.92060896877526683111757824735, 6.11535773269686048163206627505, 6.47134108718901015267226278878, 7.44905230758568324751638489480, 8.204682954127224865447567993561
