L(s) = 1 | − i·5-s − 7-s + 9-s + 2i·11-s + 2i·13-s − 25-s + i·35-s − i·45-s + 2·47-s + 49-s + 2·55-s − 63-s + 2·65-s − 2i·77-s + 81-s + ⋯ |
L(s) = 1 | − i·5-s − 7-s + 9-s + 2i·11-s + 2i·13-s − 25-s + i·35-s − i·45-s + 2·47-s + 49-s + 2·55-s − 63-s + 2·65-s − 2i·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.053759558\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053759558\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - T^{2} \) |
| 11 | \( 1 - 2iT - T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 2T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.459881325641172899016079811507, −8.831394442990313583670878376668, −7.55520614881860926426759513069, −7.02670398501714536064140783222, −6.40245484281606574642433087981, −5.18631535594364767818217546896, −4.23991862243213802322085863669, −4.10423552562935204928437904608, −2.28546716966269527939844666481, −1.50260973538314362972585029969,
0.77279827663538190267141910458, 2.66811537291098219243915336334, 3.26885473350943656766464002189, 3.92598154127960412808738640344, 5.50659459382391024198376745211, 5.96572203206005556189432966188, 6.76221389566896026746823082254, 7.57091251905719809600274605728, 8.245410857275170765804154857367, 9.188633971111568416350869876617