| L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s + 8-s − 9-s − 6i·11-s + i·12-s + (−3 − 2i)13-s + 16-s − 6i·17-s − 18-s + 6i·19-s − 6i·22-s − 6i·23-s + i·24-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 0.353·8-s − 0.333·9-s − 1.80i·11-s + 0.288i·12-s + (−0.832 − 0.554i)13-s + 0.250·16-s − 1.45i·17-s − 0.235·18-s + 1.37i·19-s − 1.27i·22-s − 1.25i·23-s + 0.204i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.225543528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.225543528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.930278877378607760390344078415, −8.369816244060926049177740186628, −7.42966807285281778340891904716, −6.49823799587528128001583628609, −5.59457226481515137352931259770, −5.13625274812012856265925250373, −4.05879312497700303639744137159, −3.22679869694037814194190322616, −2.48844208535735302867098881159, −0.61019714111851710835894311618,
1.59571430115849590502522512809, 2.32389981412864663844323002376, 3.48315945937649641266676623612, 4.68108337011844211443211702665, 4.99549780300333204117252614900, 6.38267915457396374874034488908, 6.79660075874623205189525585507, 7.56882300376417049436495550039, 8.305229389454586030783681884608, 9.515972019339228845688071230401
