L(s) = 1 | − 2-s + 4-s − 4.42·7-s − 8-s + 5.80·11-s + 6.42·13-s + 4.42·14-s + 16-s − 3.37·17-s − 19-s − 5.80·22-s − 6.42·23-s − 6.42·26-s − 4.42·28-s − 7.80·29-s + 9.05·31-s − 32-s + 3.37·34-s − 3.67·37-s + 38-s − 4.42·41-s + 1.05·43-s + 5.80·44-s + 6.42·46-s + 5.18·47-s + 12.6·49-s + 6.42·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.67·7-s − 0.353·8-s + 1.75·11-s + 1.78·13-s + 1.18·14-s + 0.250·16-s − 0.819·17-s − 0.229·19-s − 1.23·22-s − 1.34·23-s − 1.26·26-s − 0.836·28-s − 1.44·29-s + 1.62·31-s − 0.176·32-s + 0.579·34-s − 0.603·37-s + 0.162·38-s − 0.691·41-s + 0.160·43-s + 0.875·44-s + 0.947·46-s + 0.756·47-s + 1.80·49-s + 0.891·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 + 0.133T + 83T^{2} \) |
| 89 | \( 1 - 3.18T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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
−7.34259352230170570617543730828, −6.60877166144834602510749654782, −6.21470719207236599319447571225, −5.90933538391036397910953443821, −4.30933140601898424930468949964, −3.73965198960801103894794851466, −3.19552415553085885307725142898, −2.01454928092403901511776494726, −1.14527809534458174151834125722, 0,
1.14527809534458174151834125722, 2.01454928092403901511776494726, 3.19552415553085885307725142898, 3.73965198960801103894794851466, 4.30933140601898424930468949964, 5.90933538391036397910953443821, 6.21470719207236599319447571225, 6.60877166144834602510749654782, 7.34259352230170570617543730828