| 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
−13.41353931617511179086806270648, −12.20480199656926158816991073567, −11.25441262139326080499625220169, −10.34532644673397166246600244326, −10.03662884387557351429674899543, −8.907768756358817043040653179284, −6.96793873389174145189581807551, −6.11699950256766796445645161840, −4.07546276413086268434945474012, −2.96751879559100533179879741090,
0.11518966555951668735707982953, 3.20059139481748897845818745958, 5.66932000991120382186019859316, 6.22668831056760469429905055941, 7.52353204911666376812348701796, 8.466005023780266418940336345733, 9.219731131191818446465768233981, 10.85317790483718991292073060949, 12.20632999596134760061225486669, 12.72315798747598120194643799255
