L(s) = 1 | − 2.11·2-s + 3-s + 2.46·4-s − 3.85·5-s − 2.11·6-s + 0.0837·7-s − 0.975·8-s + 9-s + 8.14·10-s + 3.28·11-s + 2.46·12-s − 3.49·13-s − 0.176·14-s − 3.85·15-s − 2.86·16-s + 17-s − 2.11·18-s + 4.74·19-s − 9.49·20-s + 0.0837·21-s − 6.94·22-s + 0.819·23-s − 0.975·24-s + 9.88·25-s + 7.37·26-s + 27-s + 0.206·28-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.23·4-s − 1.72·5-s − 0.862·6-s + 0.0316·7-s − 0.344·8-s + 0.333·9-s + 2.57·10-s + 0.991·11-s + 0.710·12-s − 0.968·13-s − 0.0472·14-s − 0.996·15-s − 0.715·16-s + 0.242·17-s − 0.497·18-s + 1.08·19-s − 2.12·20-s + 0.0182·21-s − 1.48·22-s + 0.170·23-s − 0.199·24-s + 1.97·25-s + 1.44·26-s + 0.192·27-s + 0.0389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8194064260\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8194064260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 0.0837T + 7T^{2} \) |
| 11 | \( 1 - 3.28T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 - 0.819T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 0.312T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 - 7.91T + 43T^{2} \) |
| 47 | \( 1 - 9.53T + 47T^{2} \) |
| 53 | \( 1 + 1.20T + 53T^{2} \) |
| 59 | \( 1 - 1.03T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 9.27T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 6.83T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80129328966055196351542089337, −7.43334485262225503914794717704, −7.07583400921745471227628351914, −6.08109497636013026043028285967, −4.62069498819555289908499941244, −4.36046018297101480644815517940, −3.28368220008578527632909408079, −2.65845200593632024659444059237, −1.34480090177013283049065664977, −0.61312885542531801053573508446,
0.61312885542531801053573508446, 1.34480090177013283049065664977, 2.65845200593632024659444059237, 3.28368220008578527632909408079, 4.36046018297101480644815517940, 4.62069498819555289908499941244, 6.08109497636013026043028285967, 7.07583400921745471227628351914, 7.43334485262225503914794717704, 7.80129328966055196351542089337