| L(s) = 1 | + i·2-s + 0.732i·3-s − 4-s − 0.732·6-s + 4.73i·7-s − i·8-s + 2.46·9-s − 5.46·11-s − 0.732i·12-s − 5.46i·13-s − 4.73·14-s + 16-s − 5.46i·17-s + 2.46i·18-s − 6.19·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.422i·3-s − 0.5·4-s − 0.298·6-s + 1.78i·7-s − 0.353i·8-s + 0.821·9-s − 1.64·11-s − 0.211i·12-s − 1.51i·13-s − 1.26·14-s + 0.250·16-s − 1.32i·17-s + 0.580i·18-s − 1.42·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3450148628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3450148628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.732iT - 3T^{2} \) |
| 7 | \( 1 - 4.73iT - 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 5.46iT - 13T^{2} \) |
| 17 | \( 1 + 5.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 0.732T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 4.73iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 3.66iT - 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 + 0.928iT - 73T^{2} \) |
| 79 | \( 1 + 8.73T + 79T^{2} \) |
| 83 | \( 1 + 8.73iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.942173285341218760817588010198, −8.291940433820212207826445398715, −7.70646175835601840219396487785, −6.65029793566055903355546308856, −5.71592599953374055210745278461, −5.18063091829708337009660101916, −4.52191808866164364804430068844, −2.99835697077606186585073568699, −2.35536341556365986370658436055, −0.12368224523285514056220162979,
1.41900352738111328028068348418, 2.14137728201475462284399007313, 3.80047590515608099156433508570, 4.07940059263292907150033875044, 5.10136191615120178248826050008, 6.35424620082476919540345432579, 7.20234541792437132042485052317, 7.70062123395740193968349855408, 8.556231821216323877857600764886, 9.664251303445353794150741682480
