| L(s) = 1 | + (1.13 + 0.367i)2-s + (−1.22 + 1.22i)3-s + (−0.473 − 0.343i)4-s + (3.25 + 1.88i)5-s + (−1.83 + 0.940i)6-s + (−0.121 + 1.15i)7-s + (−1.80 − 2.48i)8-s + (−0.0155 − 2.99i)9-s + (2.99 + 3.32i)10-s + (−1.44 + 0.642i)11-s + (1.00 − 0.161i)12-s + (1.12 − 5.27i)13-s + (−0.561 + 1.26i)14-s + (−6.29 + 1.70i)15-s + (−0.768 − 2.36i)16-s + (−3.70 − 1.64i)17-s + ⋯ |
| L(s) = 1 | + (0.799 + 0.259i)2-s + (−0.705 + 0.708i)3-s + (−0.236 − 0.171i)4-s + (1.45 + 0.841i)5-s + (−0.748 + 0.383i)6-s + (−0.0458 + 0.436i)7-s + (−0.639 − 0.879i)8-s + (−0.00517 − 0.999i)9-s + (0.947 + 1.05i)10-s + (−0.434 + 0.193i)11-s + (0.288 − 0.0465i)12-s + (0.310 − 1.46i)13-s + (−0.150 + 0.336i)14-s + (−1.62 + 0.439i)15-s + (−0.192 − 0.591i)16-s + (−0.898 − 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11356 + 0.530063i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11356 + 0.530063i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.22 - 1.22i)T \) |
| 31 | \( 1 + (-1.59 - 5.33i)T \) |
| good | 2 | \( 1 + (-1.13 - 0.367i)T + (1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-3.25 - 1.88i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.121 - 1.15i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (1.44 - 0.642i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.12 + 5.27i)T + (-11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.70 + 1.64i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 0.440i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (1.05 - 0.765i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.285 + 0.878i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (7.98 - 4.60i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.74 + 2.47i)T + (4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (1.64 + 7.73i)T + (-39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (7.19 - 2.33i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.927 - 8.82i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (7.74 + 6.97i)T + (6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 4.86iT - 61T^{2} \) |
| 67 | \( 1 + (-1.87 + 3.25i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.89 + 0.830i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.894 - 2.00i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-1.85 + 4.16i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (3.43 + 3.81i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-2.01 - 1.46i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.65 - 6.28i)T + (29.9 + 92.2i)T^{2} \) |
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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
−14.13619118395042827710081459543, −13.37101107430871282311578448568, −12.30147181542561963706722599676, −10.70502163283678458530308323602, −10.07866826617073084012623082453, −9.083326864583451507126980728938, −6.69697637375726005450942397519, −5.74746521006105215217523391490, −5.06790325546125228658520656489, −3.12497737304880856818080558665,
1.99456303751437572293170575328, 4.48195616119776469926263029162, 5.54213362125784773341979343871, 6.56555005558881394554294103271, 8.390770709183807430398733741865, 9.533721278431197469590480590626, 11.00152336050046777521588386940, 12.09455265458980625225984319827, 13.11278073038372706373209617614, 13.51623883311932416880506426820
