| L(s) = 1 | + (−0.856 + 1.22i)2-s + (−0.960 − 1.44i)3-s + (−0.0790 − 0.217i)4-s + (−1.10 − 1.94i)5-s + (2.58 + 0.0591i)6-s + (−4.36 + 2.03i)7-s + (−2.55 − 0.683i)8-s + (−1.15 + 2.76i)9-s + (3.32 + 0.312i)10-s + (−2.25 − 2.68i)11-s + (−0.237 + 0.322i)12-s + (3.11 − 2.18i)13-s + (1.24 − 7.08i)14-s + (−1.73 + 3.46i)15-s + (3.37 − 2.83i)16-s + (−1.37 + 0.367i)17-s + ⋯ |
| L(s) = 1 | + (−0.605 + 0.865i)2-s + (−0.554 − 0.832i)3-s + (−0.0395 − 0.108i)4-s + (−0.494 − 0.869i)5-s + (1.05 + 0.0241i)6-s + (−1.65 + 0.769i)7-s + (−0.902 − 0.241i)8-s + (−0.384 + 0.923i)9-s + (1.05 + 0.0987i)10-s + (−0.678 − 0.808i)11-s + (−0.0684 + 0.0931i)12-s + (0.864 − 0.605i)13-s + (0.333 − 1.89i)14-s + (−0.448 + 0.893i)15-s + (0.844 − 0.708i)16-s + (−0.332 + 0.0891i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0335451 - 0.0866654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0335451 - 0.0866654i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.960 + 1.44i)T \) |
| 5 | \( 1 + (1.10 + 1.94i)T \) |
| good | 2 | \( 1 + (0.856 - 1.22i)T + (-0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (4.36 - 2.03i)T + (4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (2.25 + 2.68i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.11 + 2.18i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.37 - 0.367i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.30 - 0.750i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.633 - 1.35i)T + (-14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.168 - 0.957i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.44 - 0.891i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.79 + 6.69i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.670 + 0.118i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.200 + 0.0175i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.68 - 7.89i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (2.81 - 2.81i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.69 + 4.77i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.08 + 0.396i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (8.79 + 12.5i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (11.1 + 6.42i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.0920 + 0.343i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (9.66 - 1.70i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-9.30 - 6.51i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (2.09 + 3.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.561 + 6.41i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−12.72315798747598120194643799255, −12.20632999596134760061225486669, −10.85317790483718991292073060949, −9.219731131191818446465768233981, −8.466005023780266418940336345733, −7.52353204911666376812348701796, −6.22668831056760469429905055941, −5.66932000991120382186019859316, −3.20059139481748897845818745958, −0.11518966555951668735707982953,
2.96751879559100533179879741090, 4.07546276413086268434945474012, 6.11699950256766796445645161840, 6.96793873389174145189581807551, 8.907768756358817043040653179284, 10.03662884387557351429674899543, 10.34532644673397166246600244326, 11.25441262139326080499625220169, 12.20480199656926158816991073567, 13.41353931617511179086806270648
