L(s) = 1 | + 1.41·2-s − 3-s − 0.585·5-s − 1.41·6-s − 2.82·8-s + 9-s − 0.828·10-s − 0.414·11-s + 2.24·13-s + 0.585·15-s − 4.00·16-s − 2.41·17-s + 1.41·18-s + 2.58·19-s − 0.585·22-s + 1.41·23-s + 2.82·24-s − 4.65·25-s + 3.17·26-s − 27-s + 8.07·29-s + 0.828·30-s + 3·31-s + 0.414·33-s − 3.41·34-s + ⋯ |
L(s) = 1 | + 1.00·2-s − 0.577·3-s − 0.261·5-s − 0.577·6-s − 0.999·8-s + 0.333·9-s − 0.261·10-s − 0.124·11-s + 0.621·13-s + 0.151·15-s − 1.00·16-s − 0.585·17-s + 0.333·18-s + 0.593·19-s − 0.124·22-s + 0.294·23-s + 0.577·24-s − 0.931·25-s + 0.621·26-s − 0.192·27-s + 1.49·29-s + 0.151·30-s + 0.538·31-s + 0.0721·33-s − 0.585·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 6.65T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.68542676738369525551702232502, −6.62297992810949274203209241665, −6.23042952138944208166638268726, −5.50118210440411029203220875502, −4.70513986997764905516749695502, −4.29492636100223364127288075987, −3.38508879765128773443871102874, −2.65769814446075140339979762284, −1.26491025786676716303416602889, 0,
1.26491025786676716303416602889, 2.65769814446075140339979762284, 3.38508879765128773443871102874, 4.29492636100223364127288075987, 4.70513986997764905516749695502, 5.50118210440411029203220875502, 6.23042952138944208166638268726, 6.62297992810949274203209241665, 7.68542676738369525551702232502