L(s) = 1 | + 2-s − 2.62·3-s + 4-s − 1.76·5-s − 2.62·6-s − 7-s + 8-s + 3.89·9-s − 1.76·10-s + 2.86·11-s − 2.62·12-s − 14-s + 4.62·15-s + 16-s + 0.626·17-s + 3.89·18-s + 7.25·19-s − 1.76·20-s + 2.62·21-s + 2.86·22-s + 5.52·23-s − 2.62·24-s − 1.89·25-s − 2.35·27-s − 28-s + 3.76·29-s + 4.62·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s − 0.787·5-s − 1.07·6-s − 0.377·7-s + 0.353·8-s + 1.29·9-s − 0.557·10-s + 0.863·11-s − 0.758·12-s − 0.267·14-s + 1.19·15-s + 0.250·16-s + 0.151·17-s + 0.918·18-s + 1.66·19-s − 0.393·20-s + 0.573·21-s + 0.610·22-s + 1.15·23-s − 0.536·24-s − 0.379·25-s − 0.453·27-s − 0.188·28-s + 0.698·29-s + 0.844·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.210369877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.210369877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.626T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 0.626T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 + 0.509T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 5.10T + 79T^{2} \) |
| 83 | \( 1 - 0.387T + 83T^{2} \) |
| 89 | \( 1 - 8.89T + 89T^{2} \) |
| 97 | \( 1 + 7.10T + 97T^{2} \) |
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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
−11.10757363279843446708493526418, −10.17506974247704760993277180266, −9.128817925669330731053712170305, −7.63001797093701967215520971330, −6.89772780772444763321622178266, −6.04217340344211938162733859461, −5.19702182612899164574723190474, −4.27948145793271618732746558621, −3.21102121982046635593389337956, −0.987019722953690538217866446099,
0.987019722953690538217866446099, 3.21102121982046635593389337956, 4.27948145793271618732746558621, 5.19702182612899164574723190474, 6.04217340344211938162733859461, 6.89772780772444763321622178266, 7.63001797093701967215520971330, 9.128817925669330731053712170305, 10.17506974247704760993277180266, 11.10757363279843446708493526418