L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.587 + 0.809i)5-s + (−0.309 + 0.951i)6-s + 1.90i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.951 + 0.690i)11-s + (0.587 + 0.809i)12-s + (1.53 + 1.11i)14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.587 + 0.809i)5-s + (−0.309 + 0.951i)6-s + 1.90i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.951 + 0.690i)11-s + (0.587 + 0.809i)12-s + (1.53 + 1.11i)14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s − 0.999i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7844907562\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7844907562\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
good | 7 | \( 1 - 1.90iT - T^{2} \) |
| 11 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \) |
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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
−11.25728803910112672129028260698, −10.27610051670775992329844255479, −9.495467906962375405625387362495, −8.660360851086245252540820849826, −6.96876974069247418687184008523, −6.18372284999324843465077617599, −5.37361191487789018544182214407, −4.37381509637465213202840883205, −3.28274381420651989080741351027, −2.01116392648208145479120100076,
0.923675753889546379604728083777, 3.81248413234894451597106078749, 4.24580839114015138033107261502, 5.28993817523873432170738279333, 6.35931404959401955011513213856, 7.17220115185304875276317963819, 7.76439853201747349661867894972, 8.790538772543956982718231283145, 10.01188898512626109191776756953, 11.14656748063047182982342016809