| L(s) = 1 | − 1.02e3·2-s − 1.22e4·3-s + 1.04e6·4-s + 1.25e7·6-s − 9.06e8·7-s − 1.07e9·8-s − 1.03e10·9-s + 8.75e10·11-s − 1.28e10·12-s − 3.78e10·13-s + 9.27e11·14-s + 1.09e12·16-s − 1.03e13·17-s + 1.05e13·18-s − 4.47e13·19-s + 1.10e13·21-s − 8.96e13·22-s − 2.93e14·23-s + 1.31e13·24-s + 3.87e13·26-s + 2.54e14·27-s − 9.50e14·28-s − 4.56e14·29-s + 5.31e15·31-s − 1.12e15·32-s − 1.07e15·33-s + 1.05e16·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.119·3-s + 0.5·4-s + 0.0845·6-s − 1.21·7-s − 0.353·8-s − 0.985·9-s + 1.01·11-s − 0.0598·12-s − 0.0761·13-s + 0.857·14-s + 0.250·16-s − 1.24·17-s + 0.696·18-s − 1.67·19-s + 0.145·21-s − 0.719·22-s − 1.47·23-s + 0.0422·24-s + 0.0538·26-s + 0.237·27-s − 0.606·28-s − 0.201·29-s + 1.16·31-s − 0.176·32-s − 0.121·33-s + 0.877·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.2102479832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2102479832\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + 1.22e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 9.06e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.75e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.78e10T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.03e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.47e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.93e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 4.56e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.31e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.33e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.54e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.59e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.36e15T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.47e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.78e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.95e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 3.15e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.60e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 5.16e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.21e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.00e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.72e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 5.06e20T + 5.27e41T^{2} \) |
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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
−11.30097048887196223122799475811, −10.13761089718158929617653657014, −9.116564193357726586171272056387, −8.265603597617465292966324214728, −6.53525016400548429121198003217, −6.21570973424463301163526919381, −4.25555708082972646429059387951, −2.97277030419882054142181476157, −1.85403423575163878546077054850, −0.21850794754384001389475704156,
0.21850794754384001389475704156, 1.85403423575163878546077054850, 2.97277030419882054142181476157, 4.25555708082972646429059387951, 6.21570973424463301163526919381, 6.53525016400548429121198003217, 8.265603597617465292966324214728, 9.116564193357726586171272056387, 10.13761089718158929617653657014, 11.30097048887196223122799475811
