| L(s) = 1 | − i·2-s − 4-s + (1.80 − 1.93i)7-s + i·8-s − 3.87i·11-s + 1.60i·13-s + (−1.93 − 1.80i)14-s + 16-s + 8.11·17-s + 2.63i·19-s − 3.87·22-s + 5.47i·23-s + 1.60·26-s + (−1.80 + 1.93i)28-s − 5.47i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.681 − 0.732i)7-s + 0.353i·8-s − 1.16i·11-s + 0.445i·13-s + (−0.517 − 0.481i)14-s + 0.250·16-s + 1.96·17-s + 0.605i·19-s − 0.825·22-s + 1.14i·23-s + 0.314·26-s + (−0.340 + 0.366i)28-s − 1.01i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011780741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011780741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.80 + 1.93i)T \) |
| good | 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 - 1.60iT - 13T^{2} \) |
| 17 | \( 1 - 8.11T + 17T^{2} \) |
| 19 | \( 1 - 2.63iT - 19T^{2} \) |
| 23 | \( 1 - 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 + 3.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 2.26iT - 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 6.90T + 67T^{2} \) |
| 71 | \( 1 + 2.63iT - 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 8.68iT - 97T^{2} \) |
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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
−8.469779682716811678154343086974, −7.78118576727864105650735894649, −7.28577367499572498744606648605, −5.77875101479241603637731327369, −5.61702957622893723067761535856, −4.29065677719626743566731098063, −3.74915386234879738297091298350, −2.87680318727254087429251046962, −1.59180717863551160522424372429, −0.78102243317445115726269937994,
1.10403384854109550611903610918, 2.33860276417307073545031763832, 3.36873456936668321110709345846, 4.54316725227302106917326252369, 5.12832656979728311263196489823, 5.76616728274029794980606976015, 6.68118405144185844453505601346, 7.57766948770159837064468441574, 7.87888392605199905807697204666, 8.935502448673464099736855907619
