L(s) = 1 | + (1 + 1.73i)2-s + (−0.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (−0.5 − 0.866i)5-s − 1.99·6-s + (−0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (0.999 − 1.73i)10-s + (3 − 5.19i)11-s + (−1 − 1.73i)12-s − 3·13-s + (−5 + 1.73i)14-s + 0.999·15-s + (1.99 + 3.46i)16-s + (2 − 3.46i)17-s + (0.999 − 1.73i)18-s + ⋯ |
L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.288 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (−0.223 − 0.387i)5-s − 0.816·6-s + (−0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (0.904 − 1.56i)11-s + (−0.288 − 0.499i)12-s − 0.832·13-s + (−1.33 + 0.462i)14-s + 0.258·15-s + (0.499 + 0.866i)16-s + (0.485 − 0.840i)17-s + (0.235 − 0.408i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.779998 + 1.02529i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.779998 + 1.02529i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−14.36217335027837631298114730930, −13.31427756038192158740690064611, −12.09946190336053656384798194936, −11.18868243653864648732214163107, −9.458280534738678381554534457946, −8.540374159858362232788619976984, −7.13040338585399542146241657938, −5.81423333869221498209692112067, −5.17542032033890415550538412225, −3.60095122403794590604848515797,
1.83324032084037973742263210554, 3.65736222745361496227383056214, 4.77609586379963178204813101166, 6.74599157926910595495511504158, 7.62877944110981313382173921888, 9.764649106786253356832991372553, 10.50648944710682975799363096614, 11.59567067679566165484138677356, 12.42797169331384226126533638833, 13.06727659824459333800968084157