L(s) = 1 | − 1.05e4·5-s − 1.68e4·7-s + 4.87e5·11-s − 2.94e4·13-s − 2.33e6·17-s − 1.94e6·19-s − 1.87e7·23-s + 6.27e7·25-s − 1.99e7·29-s − 2.43e8·31-s + 1.77e8·35-s + 5.32e8·37-s + 8.90e8·41-s + 5.15e8·43-s + 2.38e9·47-s + 2.82e8·49-s + 3.59e9·53-s − 5.15e9·55-s + 4.83e9·59-s + 2.98e9·61-s + 3.10e8·65-s + 1.87e10·67-s + 5.13e9·71-s − 3.28e10·73-s − 8.19e9·77-s − 2.03e10·79-s + 1.70e10·83-s + ⋯ |
L(s) = 1 | − 1.51·5-s − 0.377·7-s + 0.913·11-s − 0.0219·13-s − 0.398·17-s − 0.179·19-s − 0.608·23-s + 1.28·25-s − 0.180·29-s − 1.52·31-s + 0.571·35-s + 1.26·37-s + 1.20·41-s + 0.534·43-s + 1.51·47-s + 0.142·49-s + 1.18·53-s − 1.38·55-s + 0.879·59-s + 0.452·61-s + 0.0332·65-s + 1.69·67-s + 0.337·71-s − 1.85·73-s − 0.345·77-s − 0.742·79-s + 0.474·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
good | 5 | \( 1 + 1.05e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 4.87e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.94e4T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.33e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.94e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.87e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.99e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.43e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.32e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.90e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.15e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.38e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.59e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.83e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.98e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.87e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 5.13e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.28e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.03e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.70e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.06e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 9.26e10T + 7.15e21T^{2} \) |
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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
−9.489799215609054040441788266227, −8.642848109634703128078575213345, −7.63387946105224183202334502966, −6.89121284941605057620367017602, −5.71012130317550334942519739132, −4.16072320033572692206982767457, −3.84399257037907854215826262180, −2.47296072030607086807977270049, −0.955745956902430902896537478697, 0,
0.955745956902430902896537478697, 2.47296072030607086807977270049, 3.84399257037907854215826262180, 4.16072320033572692206982767457, 5.71012130317550334942519739132, 6.89121284941605057620367017602, 7.63387946105224183202334502966, 8.642848109634703128078575213345, 9.489799215609054040441788266227