L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 4·11-s + 4·13-s + 4·14-s + 16-s − 4·17-s − 4·19-s − 20-s + 4·22-s − 8·23-s + 25-s + 4·26-s + 4·28-s + 6·29-s − 4·31-s + 32-s − 4·34-s − 4·35-s − 2·37-s − 4·38-s − 40-s − 10·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.755·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.249583633\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.249583633\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.40665317551619, −12.72912955731660, −12.12358460691484, −11.89137852626337, −11.33195389344490, −11.12395324321480, −10.59598006300286, −10.07241460700438, −9.375669014421465, −8.574045715148531, −8.383306325828700, −8.197820754998350, −7.277826422190017, −6.816139696548886, −6.396593402968976, −5.879803775593373, −5.254412375095637, −4.613101082520201, −4.303286755920876, −3.788775675131747, −3.415981373897205, −2.335267731020194, −1.874912667338656, −1.449177948895119, −0.5780992828847850,
0.5780992828847850, 1.449177948895119, 1.874912667338656, 2.335267731020194, 3.415981373897205, 3.788775675131747, 4.303286755920876, 4.613101082520201, 5.254412375095637, 5.879803775593373, 6.396593402968976, 6.816139696548886, 7.277826422190017, 8.197820754998350, 8.383306325828700, 8.574045715148531, 9.375669014421465, 10.07241460700438, 10.59598006300286, 11.12395324321480, 11.33195389344490, 11.89137852626337, 12.12358460691484, 12.72912955731660, 13.40665317551619