L(s) = 1 | + (1.20 + 0.742i)2-s + (0.819 + 1.52i)3-s + (0.896 + 1.78i)4-s + (−0.206 − 0.769i)5-s + (−0.146 + 2.44i)6-s + (−2.17 − 3.76i)7-s + (−0.248 + 2.81i)8-s + (−1.65 + 2.50i)9-s + (0.323 − 1.07i)10-s + (1.05 − 3.93i)11-s + (−1.99 + 2.83i)12-s + (−0.454 − 1.69i)13-s + (0.180 − 6.15i)14-s + (1.00 − 0.945i)15-s + (−2.39 + 3.20i)16-s + 6.68i·17-s + ⋯ |
L(s) = 1 | + (0.850 + 0.525i)2-s + (0.473 + 0.880i)3-s + (0.448 + 0.893i)4-s + (−0.0922 − 0.344i)5-s + (−0.0598 + 0.998i)6-s + (−0.822 − 1.42i)7-s + (−0.0879 + 0.996i)8-s + (−0.551 + 0.833i)9-s + (0.102 − 0.341i)10-s + (0.318 − 1.18i)11-s + (−0.575 + 0.818i)12-s + (−0.126 − 0.470i)13-s + (0.0482 − 1.64i)14-s + (0.259 − 0.244i)15-s + (−0.598 + 0.801i)16-s + 1.62i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47895 + 0.983969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47895 + 0.983969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 0.742i)T \) |
| 3 | \( 1 + (-0.819 - 1.52i)T \) |
good | 5 | \( 1 + (0.206 + 0.769i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.05 + 3.93i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.454 + 1.69i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 6.68iT - 17T^{2} \) |
| 19 | \( 1 + (0.708 + 0.708i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.88 - 2.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.06 - 3.98i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (4.94 + 2.85i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.51 + 1.51i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.36 + 2.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.60 + 2.30i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.23 - 2.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.68 + 1.68i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.00 - 0.269i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.97 - 0.528i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-8.01 + 2.14i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 8.05iT - 71T^{2} \) |
| 73 | \( 1 - 9.73iT - 73T^{2} \) |
| 79 | \( 1 + (-11.9 + 6.91i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.05 - 0.817i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 1.71T + 89T^{2} \) |
| 97 | \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.37838458945844129937571863927, −12.77366087607845957305971112630, −11.09380932163319221974365747341, −10.44618986301571955520028084572, −8.958717740724102404955489004210, −8.012903923415235566376048363644, −6.72084212213591984432471620635, −5.41822489432710945701831713207, −3.99504771358009772394264269371, −3.35527998323633432085115989534,
2.19106341210851240441535424720, 3.17211626326148265993543794902, 5.07324510598912633385392258934, 6.47446581729056576556082349034, 7.14310856036029314145022925849, 9.044161028101068368240311520837, 9.689017004848581295458202100068, 11.39762999894970801446289818461, 12.18900673071176270854364804637, 12.73582303675805961308406587382