L(s) = 1 | − 2-s + 4-s − 1.12·5-s − 0.686·7-s − 8-s + 1.12·10-s + 3.05·11-s − 3.99·13-s + 0.686·14-s + 16-s + 5.80·17-s + 8.32·19-s − 1.12·20-s − 3.05·22-s − 2.20·23-s − 3.73·25-s + 3.99·26-s − 0.686·28-s + 6.02·29-s + 8.16·31-s − 32-s − 5.80·34-s + 0.772·35-s + 4.59·37-s − 8.32·38-s + 1.12·40-s + 5.82·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.503·5-s − 0.259·7-s − 0.353·8-s + 0.355·10-s + 0.920·11-s − 1.10·13-s + 0.183·14-s + 0.250·16-s + 1.40·17-s + 1.90·19-s − 0.251·20-s − 0.651·22-s − 0.459·23-s − 0.746·25-s + 0.782·26-s − 0.129·28-s + 1.11·29-s + 1.46·31-s − 0.176·32-s − 0.994·34-s + 0.130·35-s + 0.754·37-s − 1.34·38-s + 0.177·40-s + 0.910·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363794449\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363794449\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 0.686T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 + 3.99T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 - 8.32T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 - 8.16T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 - 0.944T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 + 8.18T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 6.33T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 8.49T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 - 6.07T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.85074920079866069107167402261, −7.36188626042468557205260888189, −6.57066902072783283493694675763, −5.91137927951198106330319943778, −5.05575752875540403641433391988, −4.24839671283980084333966905542, −3.28063526915871358777355638133, −2.78521047784654782706156700269, −1.48481388395072326007553031886, −0.69125529516605066899667742817,
0.69125529516605066899667742817, 1.48481388395072326007553031886, 2.78521047784654782706156700269, 3.28063526915871358777355638133, 4.24839671283980084333966905542, 5.05575752875540403641433391988, 5.91137927951198106330319943778, 6.57066902072783283493694675763, 7.36188626042468557205260888189, 7.85074920079866069107167402261