L(s) = 1 | + 2-s − 3.05·3-s + 4-s − 1.32·5-s − 3.05·6-s − 7-s + 8-s + 6.35·9-s − 1.32·10-s − 3.05·12-s + 0.653·13-s − 14-s + 4.05·15-s + 16-s − 2.23·17-s + 6.35·18-s + 19-s − 1.32·20-s + 3.05·21-s − 3.31·23-s − 3.05·24-s − 3.23·25-s + 0.653·26-s − 10.2·27-s − 28-s − 0.248·29-s + 4.05·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.593·5-s − 1.24·6-s − 0.377·7-s + 0.353·8-s + 2.11·9-s − 0.419·10-s − 0.883·12-s + 0.181·13-s − 0.267·14-s + 1.04·15-s + 0.250·16-s − 0.543·17-s + 1.49·18-s + 0.229·19-s − 0.296·20-s + 0.667·21-s − 0.691·23-s − 0.624·24-s − 0.647·25-s + 0.128·26-s − 1.97·27-s − 0.188·28-s − 0.0461·29-s + 0.741·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 3.05T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 - 0.653T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 0.248T + 29T^{2} \) |
| 31 | \( 1 - 7.26T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 + 0.746T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 8.49T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.63389003188806535374672404329, −6.95985183303529843370675124955, −6.28461387930368549504898628071, −5.82207050473904010906292102302, −5.03164629596089941921001749123, −4.30083950446547402011017286253, −3.76635054905261108095672162991, −2.47213319003062956955066229051, −1.14307188302283604162238986513, 0,
1.14307188302283604162238986513, 2.47213319003062956955066229051, 3.76635054905261108095672162991, 4.30083950446547402011017286253, 5.03164629596089941921001749123, 5.82207050473904010906292102302, 6.28461387930368549504898628071, 6.95985183303529843370675124955, 7.63389003188806535374672404329