L(s) = 1 | + 2-s − 3.24·3-s + 4-s − 3.01·5-s − 3.24·6-s − 2.55·7-s + 8-s + 7.53·9-s − 3.01·10-s − 1.96·11-s − 3.24·12-s + 0.0961·13-s − 2.55·14-s + 9.77·15-s + 16-s − 2.89·17-s + 7.53·18-s + 6.48·19-s − 3.01·20-s + 8.27·21-s − 1.96·22-s − 2.52·23-s − 3.24·24-s + 4.06·25-s + 0.0961·26-s − 14.7·27-s − 2.55·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.87·3-s + 0.5·4-s − 1.34·5-s − 1.32·6-s − 0.963·7-s + 0.353·8-s + 2.51·9-s − 0.952·10-s − 0.591·11-s − 0.936·12-s + 0.0266·13-s − 0.681·14-s + 2.52·15-s + 0.250·16-s − 0.702·17-s + 1.77·18-s + 1.48·19-s − 0.673·20-s + 1.80·21-s − 0.418·22-s − 0.526·23-s − 0.662·24-s + 0.813·25-s + 0.0188·26-s − 2.82·27-s − 0.481·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2522231850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2522231850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 0.0961T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 9.39T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 0.0230T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
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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
−7.59931987334776216948259341713, −7.16202129904317164609001824239, −6.61297689154566457124034273674, −5.81029824113692266688798465090, −5.22896434624859375099474161063, −4.61149580321075430991483329102, −3.76198914449662464970769758140, −3.23810865172054701325569420304, −1.65714872323423031184343440623, −0.25751946227169515136723826157,
0.25751946227169515136723826157, 1.65714872323423031184343440623, 3.23810865172054701325569420304, 3.76198914449662464970769758140, 4.61149580321075430991483329102, 5.22896434624859375099474161063, 5.81029824113692266688798465090, 6.61297689154566457124034273674, 7.16202129904317164609001824239, 7.59931987334776216948259341713