| L(s) = 1 | + 503.·3-s − 3.12e3·5-s − 1.59e4·7-s + 7.60e4·9-s − 3.39e5·11-s + 2.02e6·13-s − 1.57e6·15-s − 2.45e6·17-s + 4.08e6·19-s − 8.03e6·21-s − 2.86e7·23-s + 9.76e6·25-s − 5.08e7·27-s − 9.41e6·29-s − 2.99e8·31-s − 1.70e8·33-s + 4.99e7·35-s − 4.57e8·37-s + 1.01e9·39-s + 1.83e8·41-s − 6.56e8·43-s − 2.37e8·45-s + 1.97e8·47-s − 1.72e9·49-s − 1.23e9·51-s + 5.15e9·53-s + 1.06e9·55-s + ⋯ |
| L(s) = 1 | + 1.19·3-s − 0.447·5-s − 0.359·7-s + 0.429·9-s − 0.636·11-s + 1.51·13-s − 0.534·15-s − 0.418·17-s + 0.378·19-s − 0.429·21-s − 0.928·23-s + 0.199·25-s − 0.682·27-s − 0.0852·29-s − 1.87·31-s − 0.760·33-s + 0.160·35-s − 1.08·37-s + 1.80·39-s + 0.247·41-s − 0.681·43-s − 0.192·45-s + 0.125·47-s − 0.870·49-s − 0.500·51-s + 1.69·53-s + 0.284·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 3.12e3T \) |
| good | 3 | \( 1 - 503.T + 1.77e5T^{2} \) |
| 7 | \( 1 + 1.59e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.39e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.02e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.45e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.08e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.86e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.41e6T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.99e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.57e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.83e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.56e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.97e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.15e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 6.62e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.58e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.01e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.78e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.33e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.24e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.37e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.94e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.13e11T + 7.15e21T^{2} \) |
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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
−11.55064451799008972413209465557, −10.39496465525427297207176705846, −9.049617305304588012952251711538, −8.339055487671392139091783706929, −7.26437338300738835825722499866, −5.74219042861625601784365998702, −3.94602537385463800846342186645, −3.11954479754433498865699995998, −1.76149807252644128009984901078, 0,
1.76149807252644128009984901078, 3.11954479754433498865699995998, 3.94602537385463800846342186645, 5.74219042861625601784365998702, 7.26437338300738835825722499866, 8.339055487671392139091783706929, 9.049617305304588012952251711538, 10.39496465525427297207176705846, 11.55064451799008972413209465557
