| L(s) = 1 | − 2.41e3·2-s + 3.74e6·4-s − 9.76e6·5-s − 6.35e8·7-s − 3.97e9·8-s + 2.36e10·10-s + 1.50e11·11-s − 4.38e11·13-s + 1.53e12·14-s + 1.76e12·16-s − 1.31e13·17-s + 2.30e13·19-s − 3.65e13·20-s − 3.62e14·22-s − 1.02e14·23-s + 9.53e13·25-s + 1.05e15·26-s − 2.37e15·28-s + 2.42e15·29-s + 4.77e15·31-s + 4.07e15·32-s + 3.17e16·34-s + 6.20e15·35-s − 2.00e16·37-s − 5.56e16·38-s + 3.88e16·40-s − 5.81e16·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.78·4-s − 0.447·5-s − 0.850·7-s − 1.31·8-s + 0.746·10-s + 1.74·11-s − 0.881·13-s + 1.41·14-s + 0.401·16-s − 1.57·17-s + 0.860·19-s − 0.798·20-s − 2.91·22-s − 0.513·23-s + 0.199·25-s + 1.47·26-s − 1.51·28-s + 1.07·29-s + 1.04·31-s + 0.640·32-s + 2.63·34-s + 0.380·35-s − 0.684·37-s − 1.43·38-s + 0.585·40-s − 0.676·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.4952263155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4952263155\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 2 | \( 1 + 2.41e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 6.35e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.50e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 4.38e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.31e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.30e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.02e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.42e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.77e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.00e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 5.81e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 4.75e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.37e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.87e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.80e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 2.95e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.23e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.69e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 8.57e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.76e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 5.22e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.85e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 6.77e20T + 5.27e41T^{2} \) |
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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
−11.41586485395075406210631445873, −10.07929017214923174682703796630, −9.307675874508723444423235236953, −8.438022332915854697465040224903, −7.05852121074784850908488678989, −6.48471354915892897816366653904, −4.32566673229812581174690278905, −2.84714538599221471075656815822, −1.53369838922017183346978470690, −0.43123752740041096700912603332,
0.43123752740041096700912603332, 1.53369838922017183346978470690, 2.84714538599221471075656815822, 4.32566673229812581174690278905, 6.48471354915892897816366653904, 7.05852121074784850908488678989, 8.438022332915854697465040224903, 9.307675874508723444423235236953, 10.07929017214923174682703796630, 11.41586485395075406210631445873
