L(s) = 1 | + 2·2-s − 4.22·3-s + 4·4-s + 5·5-s − 8.45·6-s + 8·8-s − 9.11·9-s + 10·10-s − 44.1·11-s − 16.9·12-s + 85.3·13-s − 21.1·15-s + 16·16-s + 107.·17-s − 18.2·18-s − 53.8·19-s + 20·20-s − 88.3·22-s − 96.1·23-s − 33.8·24-s + 25·25-s + 170.·26-s + 152.·27-s + 201.·29-s − 42.2·30-s − 90.4·31-s + 32·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.813·3-s + 0.5·4-s + 0.447·5-s − 0.575·6-s + 0.353·8-s − 0.337·9-s + 0.316·10-s − 1.21·11-s − 0.406·12-s + 1.82·13-s − 0.363·15-s + 0.250·16-s + 1.53·17-s − 0.238·18-s − 0.650·19-s + 0.223·20-s − 0.856·22-s − 0.871·23-s − 0.287·24-s + 0.200·25-s + 1.28·26-s + 1.08·27-s + 1.28·29-s − 0.257·30-s − 0.524·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.457252686\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.457252686\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 4.22T + 27T^{2} \) |
| 11 | \( 1 + 44.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 85.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 201.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 312.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 56.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 201.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 48.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 318.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 647.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 962.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 7.40T + 3.89e5T^{2} \) |
| 79 | \( 1 - 455.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 192.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 263.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.50e3T + 9.12e5T^{2} \) |
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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
−10.69489078552426354700447639019, −10.07779534490916200252762109431, −8.594461489103323620016469783626, −7.78719757834712940084801413501, −6.34176028163049461227261339162, −5.85707641203681480295962805886, −5.10217088629388004472445229627, −3.77643785604868669737165483319, −2.56771892016306920069772250815, −0.944195926306591240578956946353,
0.944195926306591240578956946353, 2.56771892016306920069772250815, 3.77643785604868669737165483319, 5.10217088629388004472445229627, 5.85707641203681480295962805886, 6.34176028163049461227261339162, 7.78719757834712940084801413501, 8.594461489103323620016469783626, 10.07779534490916200252762109431, 10.69489078552426354700447639019