| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 0.662·7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 3.57·13-s + 0.662·14-s − 15-s + 16-s − 17-s − 18-s + 5.18·19-s − 20-s − 0.662·21-s + 22-s + 7.42·23-s − 24-s + 25-s − 3.57·26-s + 27-s − 0.662·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.250·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.991·13-s + 0.176·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 1.18·19-s − 0.223·20-s − 0.144·21-s + 0.213·22-s + 1.54·23-s − 0.204·24-s + 0.200·25-s − 0.701·26-s + 0.192·27-s − 0.125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678692786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678692786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 0.662T + 7T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 - 7.42T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 0.913T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 - 5.15T + 61T^{2} \) |
| 67 | \( 1 - 5.71T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.51T + 83T^{2} \) |
| 89 | \( 1 - 0.675T + 89T^{2} \) |
| 97 | \( 1 - 7.84T + 97T^{2} \) |
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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
−8.373859448196698453038493026862, −7.49943477763318749891171051185, −6.94404610276680688362597068764, −6.23811239651166517062342425481, −5.24286703599819196462082975970, −4.42478380296312824757594256633, −3.22518153926280115490801515080, −3.06219658319286124088560118951, −1.70338685099773381614582013018, −0.77667349728611047791660233015,
0.77667349728611047791660233015, 1.70338685099773381614582013018, 3.06219658319286124088560118951, 3.22518153926280115490801515080, 4.42478380296312824757594256633, 5.24286703599819196462082975970, 6.23811239651166517062342425481, 6.94404610276680688362597068764, 7.49943477763318749891171051185, 8.373859448196698453038493026862
