| L(s) = 1 | − i·2-s + 3.25i·3-s − 4-s + 3.25·6-s + 0.0778i·7-s + i·8-s − 7.58·9-s − 4.50·11-s − 3.25i·12-s − 5.33i·13-s + 0.0778·14-s + 16-s − 7.33i·17-s + 7.58i·18-s − 19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.87i·3-s − 0.5·4-s + 1.32·6-s + 0.0294i·7-s + 0.353i·8-s − 2.52·9-s − 1.35·11-s − 0.939i·12-s − 1.47i·13-s + 0.0208·14-s + 0.250·16-s − 1.77i·17-s + 1.78i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.174712 - 0.282690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174712 - 0.282690i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.25iT - 3T^{2} \) |
| 7 | \( 1 - 0.0778iT - 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 5.33iT - 13T^{2} \) |
| 17 | \( 1 + 7.33iT - 17T^{2} \) |
| 23 | \( 1 - 3.40iT - 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 5.50iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 0.506iT - 43T^{2} \) |
| 47 | \( 1 - 5.66iT - 47T^{2} \) |
| 53 | \( 1 + 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 7.56T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 - 4.58iT - 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 5.09iT - 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 7.67iT - 97T^{2} \) |
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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
−10.03454283843838270734612256616, −9.263793454766485249278643544897, −8.421459452961983845569813549618, −7.56407974882585977326454543246, −5.67861683311281903857770880907, −5.18072649887683061693478661579, −4.44920278689761782626673028891, −3.14929335660521045419880484611, −2.81305144509178528479848249391, −0.14929890827440778306419053904,
1.58364110978100423151755258519, 2.58727923991417659482539010778, 4.19194195233440441532132139118, 5.57659150145071287807391970263, 6.20066601836138193082968009199, 6.99486071796340776832863591824, 7.63018165724741388476107830320, 8.409881865167086722914931251863, 8.945568612529122222299767869084, 10.41149668861807534945303876762
