L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 16-s + 4·17-s − 4·19-s + 20-s + 4·22-s + 6·23-s + 25-s − 4·29-s − 10·31-s − 32-s − 4·34-s − 4·37-s + 4·38-s − 40-s − 2·41-s + 12·43-s − 4·44-s − 6·46-s − 7·49-s − 50-s + 2·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 1/4·16-s + 0.970·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.25·23-s + 1/5·25-s − 0.742·29-s − 1.79·31-s − 0.176·32-s − 0.685·34-s − 0.657·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s + 1.82·43-s − 0.603·44-s − 0.884·46-s − 49-s − 0.141·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.192200137\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.192200137\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.08301947571686, −15.60874450493509, −14.92171779313211, −14.53576453969454, −13.84628714638346, −13.01770258967429, −12.76968580637946, −12.21658420369616, −11.21761303990564, −10.83020905940142, −10.46361215284205, −9.691998478076479, −9.202899237571784, −8.666612815150137, −7.856126990899947, −7.499582277109255, −6.810997794372490, −6.036064754817980, −5.402384293371917, −4.943408670795069, −3.801271321759498, −3.087641049937570, −2.308588720452664, −1.620403014090993, −0.5227807608667378,
0.5227807608667378, 1.620403014090993, 2.308588720452664, 3.087641049937570, 3.801271321759498, 4.943408670795069, 5.402384293371917, 6.036064754817980, 6.810997794372490, 7.499582277109255, 7.856126990899947, 8.666612815150137, 9.202899237571784, 9.691998478076479, 10.46361215284205, 10.83020905940142, 11.21761303990564, 12.21658420369616, 12.76968580637946, 13.01770258967429, 13.84628714638346, 14.53576453969454, 14.92171779313211, 15.60874450493509, 16.08301947571686