L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 5·7-s − 8-s + 9-s + 12-s − 6·13-s + 5·14-s + 16-s + 17-s − 18-s + 19-s − 5·21-s + 9·23-s − 24-s + 6·26-s + 27-s − 5·28-s + 6·29-s − 6·31-s − 32-s − 34-s + 36-s − 3·37-s − 38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 1.33·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.229·19-s − 1.09·21-s + 1.87·23-s − 0.204·24-s + 1.17·26-s + 0.192·27-s − 0.944·28-s + 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.493·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.16718189084262, −15.61148674641207, −14.93588124831624, −14.69767617761534, −13.81897572625673, −13.21506951990325, −12.76527223067153, −12.24442555073367, −11.78009347768958, −10.77517071977496, −10.27853331679440, −9.753076622905939, −9.404632357653852, −8.875651169430391, −8.243713958862643, −7.298479563345701, −7.001936853737260, −6.652976429106883, −5.615495434864857, −5.055687606882926, −4.042442400062344, −3.150049174755830, −2.909673842435391, −2.127658268121381, −0.9140246989361994, 0,
0.9140246989361994, 2.127658268121381, 2.909673842435391, 3.150049174755830, 4.042442400062344, 5.055687606882926, 5.615495434864857, 6.652976429106883, 7.001936853737260, 7.298479563345701, 8.243713958862643, 8.875651169430391, 9.404632357653852, 9.753076622905939, 10.27853331679440, 10.77517071977496, 11.78009347768958, 12.24442555073367, 12.76527223067153, 13.21506951990325, 13.81897572625673, 14.69767617761534, 14.93588124831624, 15.61148674641207, 16.16718189084262