L(s) = 1 | + (−0.282 + 1.38i)2-s + (−0.321 + 0.321i)3-s + (−1.83 − 0.784i)4-s + (1.77 − 1.35i)5-s + (−0.353 − 0.535i)6-s + (−1.91 − 1.91i)7-s + (1.60 − 2.32i)8-s + 2.79i·9-s + (1.37 + 2.84i)10-s − 5.65i·11-s + (0.842 − 0.338i)12-s + (−2.00 − 2.00i)13-s + (3.19 − 2.10i)14-s + (−0.134 + 1.00i)15-s + (2.77 + 2.88i)16-s + (3.89 − 3.89i)17-s + ⋯ |
L(s) = 1 | + (−0.200 + 0.979i)2-s + (−0.185 + 0.185i)3-s + (−0.919 − 0.392i)4-s + (0.794 − 0.606i)5-s + (−0.144 − 0.218i)6-s + (−0.722 − 0.722i)7-s + (0.568 − 0.822i)8-s + 0.931i·9-s + (0.435 + 0.900i)10-s − 1.70i·11-s + (0.243 − 0.0978i)12-s + (−0.555 − 0.555i)13-s + (0.852 − 0.563i)14-s + (−0.0348 + 0.259i)15-s + (0.692 + 0.721i)16-s + (0.943 − 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.983099 - 0.150731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.983099 - 0.150731i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.282 - 1.38i)T \) |
| 5 | \( 1 + (-1.77 + 1.35i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (0.321 - 0.321i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.91 + 1.91i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (2.00 + 2.00i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.89 + 3.89i)T - 17iT^{2} \) |
| 23 | \( 1 + (-3.04 + 3.04i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.56iT - 29T^{2} \) |
| 31 | \( 1 - 3.58iT - 31T^{2} \) |
| 37 | \( 1 + (-3.87 + 3.87i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.0555T + 41T^{2} \) |
| 43 | \( 1 + (1.97 - 1.97i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.74 - 4.74i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.95 + 7.95i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.34T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + (6.55 + 6.55i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.24iT - 71T^{2} \) |
| 73 | \( 1 + (3.34 + 3.34i)T + 73iT^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + (-4.60 + 4.60i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.22iT - 89T^{2} \) |
| 97 | \( 1 + (3.74 - 3.74i)T - 97iT^{2} \) |
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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
−10.89548946310661255409826683468, −10.26765174862412001645859713263, −9.378905856296349466075007017759, −8.507660800308697334397084610682, −7.56327504051467557317210380406, −6.46965279092532819978452341944, −5.48994135939119314443443886324, −4.86264048233144003801518957432, −3.21839442648448594619837931423, −0.76326385178499702357865198015,
1.79932955328298139203734671267, 2.86394855161479412958587610746, 4.17365389933414775934613371015, 5.60670152888756258314867413856, 6.58855988084356337630976892070, 7.67819587169568041375423197955, 9.253541730424178718836926069566, 9.631128175537482577923532338828, 10.25566443913834992242795856675, 11.54763350999305607304963977358