L(s) = 1 | + (−1.58 − 0.707i)3-s − 1.41·5-s + (2.23 − 1.41i)7-s + (2.00 + 2.23i)9-s + 3.16i·13-s + (2.23 + 1.00i)15-s + 2.82·17-s − 4.24i·19-s + (−4.53 + 0.654i)21-s − 6i·23-s − 2.99·25-s + (−1.58 − 4.94i)27-s − 4.47i·29-s − 5.65i·31-s + (−3.16 + 2.00i)35-s + ⋯ |
L(s) = 1 | + (−0.912 − 0.408i)3-s − 0.632·5-s + (0.845 − 0.534i)7-s + (0.666 + 0.745i)9-s + 0.877i·13-s + (0.577 + 0.258i)15-s + 0.685·17-s − 0.973i·19-s + (−0.989 + 0.142i)21-s − 1.25i·23-s − 0.599·25-s + (−0.304 − 0.952i)27-s − 0.830i·29-s − 1.01i·31-s + (−0.534 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.723888 - 0.626867i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.723888 - 0.626867i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 3.16iT - 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 13.4iT - 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 - 3.16iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 18.9iT - 97T^{2} \) |
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
−10.56136108135485758930048537403, −9.581101954135554776858480228739, −8.308172823227334916720132095260, −7.59221196503844242617342222036, −6.85389972377714019496147213440, −5.83727565922184347617439469362, −4.68223269900404205310304944905, −4.08803300965341202633061791277, −2.16933294829108335598226306306, −0.65437565394123676918291992286,
1.34033437428094932240817782453, 3.28046025678454824584829070319, 4.31475816764735676458320920034, 5.40771734392345320177182932902, 5.86579199879405800026452293482, 7.34495955953527063529227679403, 7.948882676704866144901957916563, 9.032274417406166543433046237974, 10.02013032957542389741503038788, 10.81940691474556140726269748365