L(s) = 1 | + 2-s + 4-s + 1.96·5-s − 4.72·7-s + 8-s + 1.96·10-s + 2.36·11-s − 0.742·13-s − 4.72·14-s + 16-s − 1.40·17-s − 2.69·19-s + 1.96·20-s + 2.36·22-s − 3.63·23-s − 1.13·25-s − 0.742·26-s − 4.72·28-s + 0.980·29-s + 3.58·31-s + 32-s − 1.40·34-s − 9.28·35-s + 10.6·37-s − 2.69·38-s + 1.96·40-s − 6.88·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.879·5-s − 1.78·7-s + 0.353·8-s + 0.621·10-s + 0.713·11-s − 0.205·13-s − 1.26·14-s + 0.250·16-s − 0.339·17-s − 0.617·19-s + 0.439·20-s + 0.504·22-s − 0.758·23-s − 0.226·25-s − 0.145·26-s − 0.892·28-s + 0.182·29-s + 0.644·31-s + 0.176·32-s − 0.240·34-s − 1.56·35-s + 1.74·37-s − 0.436·38-s + 0.310·40-s − 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 1.96T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 + 0.742T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 - 0.980T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 6.88T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 + 2.49T + 59T^{2} \) |
| 61 | \( 1 + 7.04T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 7.71T + 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 1.95T + 89T^{2} \) |
| 97 | \( 1 - 0.529T + 97T^{2} \) |
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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
−7.16897684495261393879638530404, −6.45140018709911269966767666911, −6.21238250181044560891387910514, −5.63197974829848806704426319584, −4.52563732665533942033413602418, −3.94571394372593372809778955046, −3.05816765419673316044942202459, −2.47877497715247089093561173255, −1.47890566537837333370781901050, 0,
1.47890566537837333370781901050, 2.47877497715247089093561173255, 3.05816765419673316044942202459, 3.94571394372593372809778955046, 4.52563732665533942033413602418, 5.63197974829848806704426319584, 6.21238250181044560891387910514, 6.45140018709911269966767666911, 7.16897684495261393879638530404