| L(s) = 1 | + 26.3·2-s + 181.·4-s − 625·5-s − 2.40e3·7-s − 8.69e3·8-s − 1.64e4·10-s + 2.24e3·11-s − 1.23e5·13-s − 6.32e4·14-s − 3.22e5·16-s + 1.87e4·17-s + 4.04e5·19-s − 1.13e5·20-s + 5.91e4·22-s − 1.75e6·23-s + 3.90e5·25-s − 3.25e6·26-s − 4.36e5·28-s − 3.94e6·29-s + 8.99e6·31-s − 4.03e6·32-s + 4.95e5·34-s + 1.50e6·35-s − 1.04e7·37-s + 1.06e7·38-s + 5.43e6·40-s − 3.02e6·41-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.355·4-s − 0.447·5-s − 0.377·7-s − 0.750·8-s − 0.520·10-s + 0.0462·11-s − 1.19·13-s − 0.440·14-s − 1.22·16-s + 0.0545·17-s + 0.712·19-s − 0.158·20-s + 0.0538·22-s − 1.30·23-s + 0.200·25-s − 1.39·26-s − 0.134·28-s − 1.03·29-s + 1.74·31-s − 0.680·32-s + 0.0635·34-s + 0.169·35-s − 0.917·37-s + 0.829·38-s + 0.335·40-s − 0.167·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.025300855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025300855\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 26.3T + 512T^{2} \) |
| 11 | \( 1 - 2.24e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.23e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.87e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.75e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.94e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.04e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.02e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.60e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.97e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.08e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.60e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.49e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.12e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.35e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.65e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.89e8T + 7.60e17T^{2} \) |
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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
−10.06714119761728645170323039803, −9.291265486685978819177547946846, −8.059195704844742061577077906875, −7.04969415679434844350810395234, −6.00478355891323906149691513245, −5.06867198490288604168787096801, −4.17710140888504093866311741419, −3.27366084986169726547234889647, −2.26616908005165407343441530810, −0.49491906700560086597567054399,
0.49491906700560086597567054399, 2.26616908005165407343441530810, 3.27366084986169726547234889647, 4.17710140888504093866311741419, 5.06867198490288604168787096801, 6.00478355891323906149691513245, 7.04969415679434844350810395234, 8.059195704844742061577077906875, 9.291265486685978819177547946846, 10.06714119761728645170323039803
