| L(s) = 1 | − 1.02e3·2-s + 1.94e5·3-s + 1.04e6·4-s + 9.76e6·5-s − 1.99e8·6-s + 9.63e8·7-s − 1.07e9·8-s + 2.74e10·9-s − 1.00e10·10-s + 6.10e10·11-s + 2.04e11·12-s + 5.25e10·13-s − 9.86e11·14-s + 1.90e12·15-s + 1.09e12·16-s − 1.13e13·17-s − 2.81e13·18-s − 4.53e13·19-s + 1.02e13·20-s + 1.87e14·21-s − 6.25e13·22-s + 2.47e14·23-s − 2.09e14·24-s + 9.53e13·25-s − 5.37e13·26-s + 3.31e15·27-s + 1.01e15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.447·5-s − 1.34·6-s + 1.28·7-s − 0.353·8-s + 2.62·9-s − 0.316·10-s + 0.710·11-s + 0.952·12-s + 0.105·13-s − 0.911·14-s + 0.851·15-s + 0.250·16-s − 1.36·17-s − 1.85·18-s − 1.69·19-s + 0.223·20-s + 2.45·21-s − 0.502·22-s + 1.24·23-s − 0.673·24-s + 0.199·25-s − 0.0747·26-s + 3.10·27-s + 0.644·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.581264138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.581264138\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.02e3T \) |
| 5 | \( 1 - 9.76e6T \) |
| good | 3 | \( 1 - 1.94e5T + 1.04e10T^{2} \) |
| 7 | \( 1 - 9.63e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 6.10e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.25e10T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.13e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.53e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.47e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.58e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.35e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.35e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 6.72e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 3.70e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.17e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.44e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.08e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.59e17T + 3.10e37T^{2} \) |
| 67 | \( 1 - 8.71e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.06e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.36e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.35e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.00e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.85e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.46e20T + 5.27e41T^{2} \) |
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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
−15.21912155159924401248132221141, −14.49413679433034969440174975376, −13.11934151912874543148122010925, −10.83923921608156252125173020419, −9.124673078026105705924920854388, −8.489474929124001929838925423219, −7.06369318748437245070050587819, −4.21010937299549460776127580880, −2.35774451691357030838514304202, −1.52849586582870041598465895635,
1.52849586582870041598465895635, 2.35774451691357030838514304202, 4.21010937299549460776127580880, 7.06369318748437245070050587819, 8.489474929124001929838925423219, 9.124673078026105705924920854388, 10.83923921608156252125173020419, 13.11934151912874543148122010925, 14.49413679433034969440174975376, 15.21912155159924401248132221141
