| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 2.24·11-s + 12-s − 5.65·13-s − 15-s + 16-s − 2.58·17-s − 18-s + 6.82·19-s − 20-s − 2.24·22-s + 3.17·23-s − 24-s + 25-s + 5.65·26-s + 27-s + 2.58·29-s + 30-s + 10.2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.676·11-s + 0.288·12-s − 1.56·13-s − 0.258·15-s + 0.250·16-s − 0.627·17-s − 0.235·18-s + 1.56·19-s − 0.223·20-s − 0.478·22-s + 0.661·23-s − 0.204·24-s + 0.200·25-s + 1.10·26-s + 0.192·27-s + 0.480·29-s + 0.182·30-s + 1.83·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367506315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367506315\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 - 0.343T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.07T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−9.575794259256383376945330298536, −8.691713581639359702128889918960, −8.000189512234428561791894612207, −7.16092719728154909945631990735, −6.71138443765284260195545084810, −5.28602139302763194320045335046, −4.38486779380176509575664886730, −3.20166362606442271447288247584, −2.36523121376459884943110542178, −0.918688926678102251627960425456,
0.918688926678102251627960425456, 2.36523121376459884943110542178, 3.20166362606442271447288247584, 4.38486779380176509575664886730, 5.28602139302763194320045335046, 6.71138443765284260195545084810, 7.16092719728154909945631990735, 8.000189512234428561791894612207, 8.691713581639359702128889918960, 9.575794259256383376945330298536
