L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (−0.309 − 0.951i)6-s − 1.90i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.951 + 0.690i)11-s + (−0.587 + 0.809i)12-s + (−1.53 + 1.11i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (−0.309 − 0.951i)6-s − 1.90i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.951 + 0.690i)11-s + (−0.587 + 0.809i)12-s + (−1.53 + 1.11i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s − 0.999i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9633640091\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9633640091\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
good | 7 | \( 1 + 1.90iT - T^{2} \) |
| 11 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
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
−10.65178954520270775939815658743, −9.920647053838673701371158469957, −9.533149096865469411332642290373, −8.094143639971374493612283476839, −7.51452233090596355162791455692, −6.82978531837306627117158375433, −4.78093643028151439514457660078, −3.84014914999646046432344837724, −2.92727719967910407026117076755, −1.75994906341256531094392674711,
1.75462904858523276402344065697, 2.83088175948821653773864595132, 4.81707902390565063278476739748, 5.65387377422654903464015565908, 6.36674009833412692200171429904, 7.79135255341540317381967523891, 8.419936306732642812229206997411, 9.026065809700060062262359292410, 9.525339031429001522540908574033, 10.60360485254817116202178101806