L(s) = 1 | − 2-s − 1.53·3-s + 4-s − 3.94·5-s + 1.53·6-s + 4.26·7-s − 8-s − 0.641·9-s + 3.94·10-s + 1.05·11-s − 1.53·12-s + 4.01·13-s − 4.26·14-s + 6.05·15-s + 16-s + 1.88·17-s + 0.641·18-s + 1.53·19-s − 3.94·20-s − 6.55·21-s − 1.05·22-s − 23-s + 1.53·24-s + 10.5·25-s − 4.01·26-s + 5.59·27-s + 4.26·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.886·3-s + 0.5·4-s − 1.76·5-s + 0.626·6-s + 1.61·7-s − 0.353·8-s − 0.213·9-s + 1.24·10-s + 0.318·11-s − 0.443·12-s + 1.11·13-s − 1.14·14-s + 1.56·15-s + 0.250·16-s + 0.456·17-s + 0.151·18-s + 0.351·19-s − 0.881·20-s − 1.43·21-s − 0.225·22-s − 0.208·23-s + 0.313·24-s + 2.11·25-s − 0.786·26-s + 1.07·27-s + 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8850617059\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8850617059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 - 4.26T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 29 | \( 1 + 5.71T + 29T^{2} \) |
| 31 | \( 1 - 1.80T + 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 + 2.93T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 9.76T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 - 6.05T + 59T^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 + 1.54T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 - 2.57T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 0.977T + 89T^{2} \) |
| 97 | \( 1 - 3.08T + 97T^{2} \) |
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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
−8.157555711431341663963528367185, −7.46670556613647809387311267215, −6.96958739792995853121036054181, −5.82357629458209964276976713738, −5.37710548219453364506040066781, −4.28265746970728024960754565602, −3.91088322733933495607465720446, −2.73200897008496547624425238713, −1.33664103747661684057544646749, −0.66094784666793181278834560310,
0.66094784666793181278834560310, 1.33664103747661684057544646749, 2.73200897008496547624425238713, 3.91088322733933495607465720446, 4.28265746970728024960754565602, 5.37710548219453364506040066781, 5.82357629458209964276976713738, 6.96958739792995853121036054181, 7.46670556613647809387311267215, 8.157555711431341663963528367185