L(s) = 1 | + 2.42·2-s + 3.86·4-s − 1.62·5-s + 2.13·7-s + 4.52·8-s − 3.93·10-s + 11-s − 5.27·13-s + 5.18·14-s + 3.22·16-s − 5.69·17-s − 3.03·19-s − 6.27·20-s + 2.42·22-s − 4.60·23-s − 2.36·25-s − 12.7·26-s + 8.27·28-s − 8.32·29-s + 2.27·31-s − 1.24·32-s − 13.7·34-s − 3.47·35-s + 0.699·37-s − 7.35·38-s − 7.34·40-s + 9.03·41-s + ⋯ |
L(s) = 1 | + 1.71·2-s + 1.93·4-s − 0.725·5-s + 0.808·7-s + 1.59·8-s − 1.24·10-s + 0.301·11-s − 1.46·13-s + 1.38·14-s + 0.805·16-s − 1.38·17-s − 0.696·19-s − 1.40·20-s + 0.516·22-s − 0.959·23-s − 0.473·25-s − 2.50·26-s + 1.56·28-s − 1.54·29-s + 0.407·31-s − 0.220·32-s − 2.36·34-s − 0.586·35-s + 0.115·37-s − 1.19·38-s − 1.16·40-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 - 0.699T + 37T^{2} \) |
| 41 | \( 1 - 9.03T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 - 4.65T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.52088933656332837838865459603, −6.87596791571759739018189325684, −6.15053676554470437664163140882, −5.36048632772081647291097833751, −4.61067631736669044119455394021, −4.24786958118217914279801657980, −3.56409763078263488785999723743, −2.37720631740297045121379954403, −1.97577356304587040320970814550, 0,
1.97577356304587040320970814550, 2.37720631740297045121379954403, 3.56409763078263488785999723743, 4.24786958118217914279801657980, 4.61067631736669044119455394021, 5.36048632772081647291097833751, 6.15053676554470437664163140882, 6.87596791571759739018189325684, 7.52088933656332837838865459603