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 | \( 1 - 7.67iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41149668861807534945303876762, −8.945568612529122222299767869084, −8.409881865167086722914931251863, −7.63018165724741388476107830320, −6.99486071796340776832863591824, −6.20066601836138193082968009199, −5.57659150145071287807391970263, −4.19194195233440441532132139118, −2.58727923991417659482539010778, −1.58364110978100423151755258519,
0.14929890827440778306419053904, 2.81305144509178528479848249391, 3.14929335660521045419880484611, 4.44920278689761782626673028891, 5.18072649887683061693478661579, 5.67861683311281903857770880907, 7.56407974882585977326454543246, 8.421459452961983845569813549618, 9.263793454766485249278643544897, 10.03454283843838270734612256616