L(s) = 1 | − 3-s − 3.24·5-s − 3.21·7-s + 9-s + 2.82·11-s − 5.85·13-s + 3.24·15-s + 1.86·17-s + 5.37·19-s + 3.21·21-s + 4.21·23-s + 5.51·25-s − 27-s + 3.33·29-s + 1.27·31-s − 2.82·33-s + 10.4·35-s − 2.06·37-s + 5.85·39-s + 2.10·41-s + 6.96·43-s − 3.24·45-s − 6.68·47-s + 3.30·49-s − 1.86·51-s + 9.34·53-s − 9.14·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.45·5-s − 1.21·7-s + 0.333·9-s + 0.850·11-s − 1.62·13-s + 0.837·15-s + 0.451·17-s + 1.23·19-s + 0.700·21-s + 0.877·23-s + 1.10·25-s − 0.192·27-s + 0.618·29-s + 0.228·31-s − 0.491·33-s + 1.75·35-s − 0.339·37-s + 0.936·39-s + 0.329·41-s + 1.06·43-s − 0.483·45-s − 0.975·47-s + 0.472·49-s − 0.260·51-s + 1.28·53-s − 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 + 2.06T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 + 5.91T + 89T^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
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
−7.85526946803841754792531138452, −7.19371947921875298803968996527, −6.88169182171200331354819980607, −5.88534184852384024657623811823, −4.98472078179642878453907895383, −4.27363998691511720042420701841, −3.42677472917950704605935964772, −2.78604943467038034466762795978, −1.01087658527285487659632088287, 0,
1.01087658527285487659632088287, 2.78604943467038034466762795978, 3.42677472917950704605935964772, 4.27363998691511720042420701841, 4.98472078179642878453907895383, 5.88534184852384024657623811823, 6.88169182171200331354819980607, 7.19371947921875298803968996527, 7.85526946803841754792531138452