| L(s) = 1 | + 2.31e7·5-s + 1.20e9·7-s + 9.89e9·11-s + 6.85e10·13-s + 1.51e12·17-s + 2.41e12·19-s + 9.58e13·23-s + 5.90e13·25-s + 2.51e15·29-s + 1.04e15·31-s + 2.77e16·35-s + 1.41e16·37-s + 1.06e17·41-s + 1.20e17·43-s − 1.55e17·47-s + 8.83e17·49-s − 1.62e18·53-s + 2.29e17·55-s − 2.38e18·59-s + 7.64e18·61-s + 1.58e18·65-s − 2.19e19·67-s + 3.93e19·71-s + 1.39e19·73-s + 1.18e19·77-s − 1.45e20·79-s + 2.58e19·83-s + ⋯ |
| L(s) = 1 | + 1.06·5-s + 1.60·7-s + 0.115·11-s + 0.137·13-s + 0.182·17-s + 0.0902·19-s + 0.482·23-s + 0.123·25-s + 1.11·29-s + 0.228·31-s + 1.70·35-s + 0.485·37-s + 1.23·41-s + 0.848·43-s − 0.432·47-s + 1.58·49-s − 1.27·53-s + 0.121·55-s − 0.607·59-s + 1.37·61-s + 0.146·65-s − 1.47·67-s + 1.43·71-s + 0.380·73-s + 0.184·77-s − 1.72·79-s + 0.182·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(4.289806796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.289806796\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.31e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 1.20e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 9.89e9T + 7.40e21T^{2} \) |
| 13 | \( 1 - 6.85e10T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.51e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.41e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 9.58e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.51e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.04e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.41e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.06e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.20e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.55e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.62e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.38e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 7.64e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.19e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.93e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.39e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.45e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.58e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 5.08e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 2.34e20T + 5.27e41T^{2} \) |
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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
−10.74556486746349621454427558874, −9.653439078237852574534731459381, −8.568224359397624493140925072369, −7.58846051831152223108907473753, −6.22342740262032528614483414276, −5.24594608542607403078802584273, −4.34300676407540939220518743834, −2.70042048403474874659360880733, −1.71027212473597410125826263804, −0.942712180828101604425581166132,
0.942712180828101604425581166132, 1.71027212473597410125826263804, 2.70042048403474874659360880733, 4.34300676407540939220518743834, 5.24594608542607403078802584273, 6.22342740262032528614483414276, 7.58846051831152223108907473753, 8.568224359397624493140925072369, 9.653439078237852574534731459381, 10.74556486746349621454427558874
