L(s) = 1 | − 2-s − 3-s + 4-s + 1.02·5-s + 6-s − 4.96·7-s − 8-s + 9-s − 1.02·10-s + 4.09·11-s − 12-s + 13-s + 4.96·14-s − 1.02·15-s + 16-s + 0.459·17-s − 18-s + 2.05·19-s + 1.02·20-s + 4.96·21-s − 4.09·22-s + 7.97·23-s + 24-s − 3.94·25-s − 26-s − 27-s − 4.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.459·5-s + 0.408·6-s − 1.87·7-s − 0.353·8-s + 0.333·9-s − 0.325·10-s + 1.23·11-s − 0.288·12-s + 0.277·13-s + 1.32·14-s − 0.265·15-s + 0.250·16-s + 0.111·17-s − 0.235·18-s + 0.472·19-s + 0.229·20-s + 1.08·21-s − 0.873·22-s + 1.66·23-s + 0.204·24-s − 0.788·25-s − 0.196·26-s − 0.192·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9938019074\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9938019074\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 1.02T + 5T^{2} \) |
| 7 | \( 1 + 4.96T + 7T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 17 | \( 1 - 0.459T + 17T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 - 5.24T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 1.59T + 47T^{2} \) |
| 53 | \( 1 - 5.49T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 1.92T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 1.18T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.65590882270170333747196409020, −6.89365930195685392462965021326, −6.65429375153886093520204225252, −5.88515269376736466081350802280, −5.40810089713008968092989284812, −4.08256593133987888582592237601, −3.45292860816252304192374862799, −2.65700814352771305134155395737, −1.47406441777318208127510455585, −0.59713687676140533316405372415,
0.59713687676140533316405372415, 1.47406441777318208127510455585, 2.65700814352771305134155395737, 3.45292860816252304192374862799, 4.08256593133987888582592237601, 5.40810089713008968092989284812, 5.88515269376736466081350802280, 6.65429375153886093520204225252, 6.89365930195685392462965021326, 7.65590882270170333747196409020