L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 0.449·11-s − 2.44·13-s + 15-s + 1.44·17-s − 7.34·19-s − 21-s + 23-s + 25-s − 27-s − 4.55·29-s + 7.89·31-s − 0.449·33-s − 35-s − 0.101·37-s + 2.44·39-s − 5.44·41-s + 11.7·43-s − 45-s − 7.34·47-s − 6·49-s − 1.44·51-s − 10.3·53-s − 0.449·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.135·11-s − 0.679·13-s + 0.258·15-s + 0.351·17-s − 1.68·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.845·29-s + 1.41·31-s − 0.0782·33-s − 0.169·35-s − 0.0166·37-s + 0.392·39-s − 0.851·41-s + 1.79·43-s − 0.149·45-s − 1.07·47-s − 0.857·49-s − 0.202·51-s − 1.42·53-s − 0.0606·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 29 | \( 1 + 4.55T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 + 0.101T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 - 5.79T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 4.89T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.206144926724832345414588024891, −8.256140839811304839536199791019, −7.59447075129003152112849114165, −6.66062457943290370015109434323, −5.92364808452699094232804809842, −4.79455298321455147735280183804, −4.27734680475250822715178961306, −2.96577361511937213992956049958, −1.63255489052141890457165172191, 0,
1.63255489052141890457165172191, 2.96577361511937213992956049958, 4.27734680475250822715178961306, 4.79455298321455147735280183804, 5.92364808452699094232804809842, 6.66062457943290370015109434323, 7.59447075129003152112849114165, 8.256140839811304839536199791019, 9.206144926724832345414588024891