L(s) = 1 | + (1.31 + 1.13i)3-s + (−0.965 + 1.67i)5-s + (−1.93 + 1.80i)7-s + (0.441 + 2.96i)9-s + (1.10 − 0.639i)11-s + (−2.52 − 1.45i)13-s + (−3.15 + 1.10i)15-s + 0.475·17-s + 6.10i·19-s + (−4.57 + 0.187i)21-s + (−6.51 − 3.75i)23-s + (0.635 + 1.09i)25-s + (−2.77 + 4.39i)27-s + (3.76 − 2.17i)29-s + (4.21 + 2.43i)31-s + ⋯ |
L(s) = 1 | + (0.757 + 0.652i)3-s + (−0.431 + 0.747i)5-s + (−0.729 + 0.683i)7-s + (0.147 + 0.989i)9-s + (0.333 − 0.192i)11-s + (−0.701 − 0.404i)13-s + (−0.815 + 0.284i)15-s + 0.115·17-s + 1.40i·19-s + (−0.999 + 0.0410i)21-s + (−1.35 − 0.783i)23-s + (0.127 + 0.219i)25-s + (−0.534 + 0.845i)27-s + (0.699 − 0.403i)29-s + (0.757 + 0.437i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.668481 + 1.19576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.668481 + 1.19576i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.31 - 1.13i)T \) |
| 7 | \( 1 + (1.93 - 1.80i)T \) |
good | 5 | \( 1 + (0.965 - 1.67i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.10 + 0.639i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.52 + 1.45i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.475T + 17T^{2} \) |
| 19 | \( 1 - 6.10iT - 19T^{2} \) |
| 23 | \( 1 + (6.51 + 3.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.76 + 2.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.21 - 2.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 + (-1.16 + 2.01i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.63 - 8.03i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.00 - 6.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (-1.74 + 3.01i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.26 + 2.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.602 + 1.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 - 1.84iT - 73T^{2} \) |
| 79 | \( 1 + (-8.54 - 14.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.225 - 0.390i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + (-9.22 + 5.32i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.06630229128038335536751131257, −10.04999590019340511304481464389, −9.706472304844400067626107768337, −8.454485213652964437115447077318, −7.85829201517844320867247402225, −6.65571812201978197434486709420, −5.64873976836042710679097601801, −4.25987889593754315027704697392, −3.29436654084346658191082044970, −2.42107822036962529333798970257,
0.74713725194569014962421647639, 2.42429842241865517589182185379, 3.75537386709748550251266095569, 4.63956451194953419930279263091, 6.24530835970607644933331263545, 7.13684143828177558748815701105, 7.82098165492079414302121698048, 8.901706870470978621070563735956, 9.479828478462381915314584678503, 10.46465803471398352187006985322