L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 4·11-s − 6·13-s + 14-s + 16-s − 4·17-s − 20-s − 4·22-s + 25-s + 6·26-s − 28-s − 6·29-s + 4·31-s − 32-s + 4·34-s + 35-s − 8·37-s + 40-s + 10·41-s − 2·43-s + 4·44-s − 10·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s + 0.169·35-s − 1.31·37-s + 0.158·40-s + 1.56·41-s − 0.304·43-s + 0.603·44-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3068985411\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3068985411\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.87771862554412, −12.33320557645780, −11.81134179167069, −11.70509728058732, −11.10227530678418, −10.51750034373098, −10.08344984473999, −9.641734030660817, −9.104481203907213, −8.881478277135992, −8.329672678258451, −7.613521353891370, −7.180481635144757, −7.045992773113225, −6.275959073150868, −5.963064272707726, −5.165648602601452, −4.566823090903944, −4.199997682425139, −3.485237606775974, −2.972044424121895, −2.298451735468230, −1.812250653301066, −1.059344481462305, −0.1862509313835156,
0.1862509313835156, 1.059344481462305, 1.812250653301066, 2.298451735468230, 2.972044424121895, 3.485237606775974, 4.199997682425139, 4.566823090903944, 5.165648602601452, 5.963064272707726, 6.275959073150868, 7.045992773113225, 7.180481635144757, 7.613521353891370, 8.329672678258451, 8.881478277135992, 9.104481203907213, 9.641734030660817, 10.08344984473999, 10.51750034373098, 11.10227530678418, 11.70509728058732, 11.81134179167069, 12.33320557645780, 12.87771862554412