L(s) = 1 | − 2.70·2-s − 2.88·3-s + 5.34·4-s − 1.42·5-s + 7.83·6-s − 3.83·7-s − 9.05·8-s + 5.34·9-s + 3.87·10-s − 1.77·11-s − 15.4·12-s − 2.78·13-s + 10.3·14-s + 4.12·15-s + 13.8·16-s + 5.22·17-s − 14.4·18-s + 1.32·19-s − 7.63·20-s + 11.0·21-s + 4.80·22-s − 7.98·23-s + 26.1·24-s − 2.95·25-s + 7.54·26-s − 6.78·27-s − 20.4·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s − 1.66·3-s + 2.67·4-s − 0.638·5-s + 3.19·6-s − 1.44·7-s − 3.20·8-s + 1.78·9-s + 1.22·10-s − 0.534·11-s − 4.45·12-s − 0.772·13-s + 2.77·14-s + 1.06·15-s + 3.46·16-s + 1.26·17-s − 3.41·18-s + 0.303·19-s − 1.70·20-s + 2.41·21-s + 1.02·22-s − 1.66·23-s + 5.34·24-s − 0.591·25-s + 1.47·26-s − 1.30·27-s − 3.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1542781111\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1542781111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 8011 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 - 4.90T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 - 5.91T + 53T^{2} \) |
| 59 | \( 1 + 0.635T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 + 2.67T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 - 9.99T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.73282290453000120451710915243, −7.34709261283220644848194737078, −6.38682547473726402314589594697, −6.22240946352607067550758016419, −5.47412525475847397464199021306, −4.33295301665560279926312217078, −3.21658919359278242308789093800, −2.41125918514363925204713296531, −1.02496770993299840523031749147, −0.35806568112831599652756511450,
0.35806568112831599652756511450, 1.02496770993299840523031749147, 2.41125918514363925204713296531, 3.21658919359278242308789093800, 4.33295301665560279926312217078, 5.47412525475847397464199021306, 6.22240946352607067550758016419, 6.38682547473726402314589594697, 7.34709261283220644848194737078, 7.73282290453000120451710915243