L(s) = 1 | − 0.107·3-s + 0.451·5-s + 2.31·7-s − 2.98·9-s + 2.84·11-s + 2.31·13-s − 0.0485·15-s + 0.174·17-s − 2.11·19-s − 0.248·21-s + 0.730·23-s − 4.79·25-s + 0.642·27-s − 7.10·29-s − 10.1·31-s − 0.305·33-s + 1.04·35-s − 2.63·37-s − 0.248·39-s − 7.51·41-s − 0.351·43-s − 1.35·45-s − 6.18·47-s − 1.63·49-s − 0.0187·51-s − 0.440·53-s + 1.28·55-s + ⋯ |
L(s) = 1 | − 0.0619·3-s + 0.202·5-s + 0.875·7-s − 0.996·9-s + 0.857·11-s + 0.641·13-s − 0.0125·15-s + 0.0424·17-s − 0.484·19-s − 0.0542·21-s + 0.152·23-s − 0.959·25-s + 0.123·27-s − 1.31·29-s − 1.81·31-s − 0.0531·33-s + 0.176·35-s − 0.432·37-s − 0.0397·39-s − 1.17·41-s − 0.0535·43-s − 0.201·45-s − 0.901·47-s − 0.232·49-s − 0.00262·51-s − 0.0605·53-s + 0.173·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 0.107T + 3T^{2} \) |
| 5 | \( 1 - 0.451T + 5T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 - 0.174T + 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 23 | \( 1 - 0.730T + 23T^{2} \) |
| 29 | \( 1 + 7.10T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 7.51T + 41T^{2} \) |
| 43 | \( 1 + 0.351T + 43T^{2} \) |
| 47 | \( 1 + 6.18T + 47T^{2} \) |
| 53 | \( 1 + 0.440T + 53T^{2} \) |
| 59 | \( 1 + 1.57T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.78T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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
−7.914422450998091835442990077705, −6.96003160392955576650800905175, −6.25308687301961274689345013913, −5.55067627032629291119821818081, −4.99791215522874238482907332487, −3.89703604187509157837270480876, −3.41444921120647940298356414113, −2.09509530110269215155165051449, −1.51628467075664754916822197814, 0,
1.51628467075664754916822197814, 2.09509530110269215155165051449, 3.41444921120647940298356414113, 3.89703604187509157837270480876, 4.99791215522874238482907332487, 5.55067627032629291119821818081, 6.25308687301961274689345013913, 6.96003160392955576650800905175, 7.914422450998091835442990077705