| L(s) = 1 | − i·2-s + i·3-s + 4-s + 6-s + 4.47i·7-s − 3i·8-s − 9-s − 1.23·11-s + i·12-s + 5.61i·13-s + 4.47·14-s − 16-s − 3.85i·17-s + i·18-s − 1.23·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + 0.408·6-s + 1.69i·7-s − 1.06i·8-s − 0.333·9-s − 0.372·11-s + 0.288i·12-s + 1.55i·13-s + 1.19·14-s − 0.250·16-s − 0.934i·17-s + 0.235i·18-s − 0.283·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526029918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526029918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 5.61iT - 13T^{2} \) |
| 17 | \( 1 + 3.85iT - 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 - 3.09iT - 37T^{2} \) |
| 41 | \( 1 + 3.61T + 41T^{2} \) |
| 43 | \( 1 - 7.70iT - 43T^{2} \) |
| 47 | \( 1 - 0.763iT - 47T^{2} \) |
| 53 | \( 1 - 3.61iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 0.763iT - 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 8.09iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 5.38T + 89T^{2} \) |
| 97 | \( 1 + 2.14iT - 97T^{2} \) |
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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
−9.366306936864327370292783768255, −9.085415799405016429820418210735, −7.989624940368910801341217537465, −6.96118168188371462012632268636, −6.15426383743042820583375723085, −5.35216840385181717081380944895, −4.42431000597508622487049298332, −3.30959112254930555157437410555, −2.51351140811126869230488384090, −1.73598547744555924589522521570,
0.51086684994932886118052218610, 1.85584500361446718580802985438, 3.06369048533634157723613553442, 4.06169395281973513419017137567, 5.26004671582219341723454394789, 5.98864905535765283405711352092, 6.85336717989599064275562177002, 7.37666483402319331323341725819, 8.033322906077083722415177073716, 8.538711363551930427398033324360
