L(s) = 1 | + 2.34·2-s + 3.18·3-s + 3.49·4-s − 5-s + 7.46·6-s − 7-s + 3.50·8-s + 7.13·9-s − 2.34·10-s + 4.62·11-s + 11.1·12-s − 6.35·13-s − 2.34·14-s − 3.18·15-s + 1.23·16-s + 7.67·17-s + 16.7·18-s + 1.74·19-s − 3.49·20-s − 3.18·21-s + 10.8·22-s − 2.85·23-s + 11.1·24-s + 25-s − 14.9·26-s + 13.1·27-s − 3.49·28-s + ⋯ |
L(s) = 1 | + 1.65·2-s + 1.83·3-s + 1.74·4-s − 0.447·5-s + 3.04·6-s − 0.377·7-s + 1.24·8-s + 2.37·9-s − 0.741·10-s + 1.39·11-s + 3.21·12-s − 1.76·13-s − 0.626·14-s − 0.822·15-s + 0.308·16-s + 1.86·17-s + 3.94·18-s + 0.399·19-s − 0.781·20-s − 0.694·21-s + 2.31·22-s − 0.595·23-s + 2.28·24-s + 0.200·25-s − 2.92·26-s + 2.53·27-s − 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.02125752\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.02125752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 - 3.18T + 3T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 6.35T + 13T^{2} \) |
| 17 | \( 1 - 7.67T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 0.436T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 + 6.26T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 + 5.81T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 6.19T + 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 2.15T + 89T^{2} \) |
| 97 | \( 1 + 3.13T + 97T^{2} \) |
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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
−7.63458158762553318357459461785, −7.19257255995204025987104458612, −6.50121248919038818420535392258, −5.60883618654143818560261789398, −4.60335211153139463456907427241, −4.21316876641240898321470598573, −3.41731126221744825148621440260, −3.03762909201212606674709687218, −2.34109178331675609911255794677, −1.31035374509227178640860612546,
1.31035374509227178640860612546, 2.34109178331675609911255794677, 3.03762909201212606674709687218, 3.41731126221744825148621440260, 4.21316876641240898321470598573, 4.60335211153139463456907427241, 5.60883618654143818560261789398, 6.50121248919038818420535392258, 7.19257255995204025987104458612, 7.63458158762553318357459461785