L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s − 3·9-s + 10-s − 4·11-s − 2·13-s − 14-s + 16-s − 6·17-s − 3·18-s − 4·19-s + 20-s − 4·22-s − 23-s + 25-s − 2·26-s − 28-s − 2·29-s + 4·31-s + 32-s − 6·34-s − 35-s − 3·36-s − 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s − 1/2·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.835681996359007970960697824366, −8.324955753884026016713284121947, −7.16815704742513423321802206505, −6.47201964357881565694892666152, −5.61122614672586450395636659363, −4.98749147127240056273145137117, −3.95133138081903200049883415413, −2.70758365609729974014748054934, −2.23446055302682209251426985281, 0,
2.23446055302682209251426985281, 2.70758365609729974014748054934, 3.95133138081903200049883415413, 4.98749147127240056273145137117, 5.61122614672586450395636659363, 6.47201964357881565694892666152, 7.16815704742513423321802206505, 8.324955753884026016713284121947, 8.835681996359007970960697824366