| L(s) = 1 | + (0.978 − 0.207i)2-s + (0.200 + 1.72i)3-s + (0.913 − 0.406i)4-s + (1.75 + 1.38i)5-s + (0.554 + 1.64i)6-s + (0.746 + 1.29i)7-s + (0.809 − 0.587i)8-s + (−2.91 + 0.691i)9-s + (2.00 + 0.993i)10-s + (0.796 − 0.169i)11-s + (0.883 + 1.48i)12-s + (−4.01 − 0.852i)13-s + (0.998 + 1.10i)14-s + (−2.03 + 3.29i)15-s + (0.669 − 0.743i)16-s + (0.388 − 0.282i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 0.147i)2-s + (0.115 + 0.993i)3-s + (0.456 − 0.203i)4-s + (0.783 + 0.620i)5-s + (0.226 + 0.669i)6-s + (0.282 + 0.488i)7-s + (0.286 − 0.207i)8-s + (−0.973 + 0.230i)9-s + (0.633 + 0.314i)10-s + (0.240 − 0.0510i)11-s + (0.254 + 0.430i)12-s + (−1.11 − 0.236i)13-s + (0.266 + 0.296i)14-s + (−0.525 + 0.850i)15-s + (0.167 − 0.185i)16-s + (0.0942 − 0.0685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02556 + 1.23998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02556 + 1.23998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 + 0.207i)T \) |
| 3 | \( 1 + (-0.200 - 1.72i)T \) |
| 5 | \( 1 + (-1.75 - 1.38i)T \) |
| good | 7 | \( 1 + (-0.746 - 1.29i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.796 + 0.169i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (4.01 + 0.852i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.388 + 0.282i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-3.79 + 2.75i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.41 + 3.79i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.142 + 1.36i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.328 - 3.12i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.63 - 5.02i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-8.51 - 1.81i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (0.371 + 0.643i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.405 + 3.85i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (3.18 + 2.31i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (6.62 + 1.40i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-10.5 + 2.24i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.270 + 2.57i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (10.7 + 7.82i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.25 - 10.0i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.491 + 4.67i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (15.0 + 6.71i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.12 + 6.52i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.0559 + 0.532i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−11.28873551095753215645995615306, −10.25046391217781036762440531739, −9.772693783482112797254584194491, −8.767283573184183011774800365595, −7.45684985192611390685782972905, −6.25644906215745260811505534726, −5.36853364787889708630707249194, −4.58076703795852387696780869783, −3.16416601287426928641789791991, −2.33345159151274160518780639596,
1.39388221839902724123245096453, 2.57887877300024417704869706473, 4.16201475540800126117106608079, 5.39783258845662390670016071959, 6.08423568801623972466377609061, 7.29289307892185254477202625325, 7.83095563752508411312724313201, 9.106927740239459687534274221673, 9.968700089455070042006435413774, 11.25528317665421344179325945030
