| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 7-s + 8-s + 9-s − 3·10-s − 5·11-s + 12-s − 14-s − 3·15-s + 16-s + 18-s + 6·19-s − 3·20-s − 21-s − 5·22-s + 2·23-s + 24-s + 4·25-s + 27-s − 28-s + 9·29-s − 3·30-s + 3·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.50·11-s + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.670·20-s − 0.218·21-s − 1.06·22-s + 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.547·30-s + 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−16.19495566706007, −15.91193673818095, −15.40387980834494, −15.23849008271074, −14.32056817301614, −13.60776658106121, −13.51505965353980, −12.55465488441039, −12.17141857537982, −11.72700963141058, −10.90522488694232, −10.39271047566692, −9.827521323326749, −8.908059373833136, −8.177535428266084, −7.862952837630769, −7.171451098756134, −6.710785662223461, −5.641986465241105, −4.989144416539782, −4.452426041944807, −3.568618824594263, −3.061754991577573, −2.570736027190322, −1.219366133849524, 0,
1.219366133849524, 2.570736027190322, 3.061754991577573, 3.568618824594263, 4.452426041944807, 4.989144416539782, 5.641986465241105, 6.710785662223461, 7.171451098756134, 7.862952837630769, 8.177535428266084, 8.908059373833136, 9.827521323326749, 10.39271047566692, 10.90522488694232, 11.72700963141058, 12.17141857537982, 12.55465488441039, 13.51505965353980, 13.60776658106121, 14.32056817301614, 15.23849008271074, 15.40387980834494, 15.91193673818095, 16.19495566706007
