| L(s) = 1 | + (−2 + i)5-s − i·7-s + 11-s + i·17-s − 19-s + (3 − 4i)25-s − 29-s − 31-s + (1 + 2i)35-s − i·37-s + 6i·43-s + 8i·47-s + 6·49-s − 9i·53-s + (−2 + i)55-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.447i)5-s − 0.377i·7-s + 0.301·11-s + 0.242i·17-s − 0.229·19-s + (0.600 − 0.800i)25-s − 0.185·29-s − 0.179·31-s + (0.169 + 0.338i)35-s − 0.164i·37-s + 0.914i·43-s + 1.16i·47-s + 0.857·49-s − 1.23i·53-s + (−0.269 + 0.134i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8165014111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8165014111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.589848963081571756047539558384, −7.892244458135302446353270382093, −7.27866468178209409860307934786, −6.60852988644062580249078422532, −5.84383432462469711236445701703, −4.75895185078397810674925630929, −4.06550093918151576739486892963, −3.38428433933015968593695670739, −2.41693139895396653084138123394, −1.09345759961013383590024521128,
0.26850257138799107860295856656, 1.54815744790336418410542536875, 2.72704416987439641473355382484, 3.67732394708881986783121408503, 4.36356271220067187678207384617, 5.18854353668630275809302840916, 5.92780326191208509714060750010, 6.91465305387354054356980180088, 7.48508418246425678002759417883, 8.305066141759858781856569625588
