L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + 1.00·6-s + (2.73 + 2.73i)7-s + (−0.707 − 0.707i)8-s − 1.99i·9-s + (0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + 3.87·14-s − 1.00·16-s + (2.73 + 2.73i)17-s + (−1.41 − 1.41i)18-s + (3.87 + 2i)19-s + 3.87i·21-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + 0.408·6-s + (1.03 + 1.03i)7-s + (−0.250 − 0.250i)8-s − 0.666i·9-s + (0.204 − 0.204i)12-s + (−0.196 − 0.196i)13-s + 1.03·14-s − 0.250·16-s + (0.664 + 0.664i)17-s + (−0.333 − 0.333i)18-s + (0.888 + 0.458i)19-s + 0.845i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.65984 - 0.327275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65984 - 0.327275i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.87 - 2i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.73 - 2.73i)T + 7iT^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.73 - 2.73i)T + 17iT^{2} \) |
| 23 | \( 1 + (-2.73 + 2.73i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.74iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (-5.47 - 5.47i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.36 + 6.36i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + (-9.19 + 9.19i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.74iT - 71T^{2} \) |
| 73 | \( 1 + (2.73 - 2.73i)T - 73iT^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (-5.47 + 5.47i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + (5.65 - 5.65i)T - 97iT^{2} \) |
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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
−9.940685575527662995321015117835, −9.302342374583890250304906638592, −8.471709458050639537094795308196, −7.68263473303682490572931028081, −6.29441291265187031752479042568, −5.48227035666506136199678850284, −4.66082503068597554460270938312, −3.58250111055890564694154404139, −2.68439013008207963646190361738, −1.41789228422973751547715776625,
1.32880128315761687337246874936, 2.69846310951555478837819754427, 3.90321886724595071014189829743, 4.91253296142835821676562999381, 5.51785758916637042498600395537, 7.14817629716852741040783357060, 7.32970462686062466009125965218, 8.085311002318164396478730756144, 9.029371095700661723825349753262, 10.08673075329966318444661538380