L(s) = 1 | − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.220 − 2.22i)5-s + (0.707 + 0.707i)6-s + (−1.18 + 1.18i)7-s + i·8-s − 1.00i·9-s + (−2.22 + 0.220i)10-s − 4.79i·11-s + (0.707 − 0.707i)12-s + 4.44i·13-s + (1.18 + 1.18i)14-s + (1.72 + 1.41i)15-s + 16-s − 1.30·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.0987 − 0.995i)5-s + (0.288 + 0.288i)6-s + (−0.448 + 0.448i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.703 + 0.0698i)10-s − 1.44i·11-s + (0.204 − 0.204i)12-s + 1.23i·13-s + (0.316 + 0.316i)14-s + (0.446 + 0.365i)15-s + 0.250·16-s − 0.315·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0385 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0385 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2943012699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2943012699\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.220 + 2.22i)T \) |
| 37 | \( 1 + (-3.31 - 5.10i)T \) |
good | 7 | \( 1 + (1.18 - 1.18i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.79iT - 11T^{2} \) |
| 13 | \( 1 - 4.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + (4.46 - 4.46i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.933iT - 23T^{2} \) |
| 29 | \( 1 + (3.66 + 3.66i)T + 29iT^{2} \) |
| 31 | \( 1 + (-4.54 + 4.54i)T - 31iT^{2} \) |
| 41 | \( 1 - 7.79iT - 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 + (6.81 - 6.81i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.08 - 5.08i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.87 - 3.87i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.36 - 6.36i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.639 - 0.639i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + (1.06 - 1.06i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.21 - 2.21i)T - 79iT^{2} \) |
| 83 | \( 1 + (6.32 + 6.32i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.11 - 3.11i)T + 89iT^{2} \) |
| 97 | \( 1 - 0.0235T + 97T^{2} \) |
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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
−9.968420932719543865194099691848, −9.305968200014201470100772936932, −8.663004711101327263553704815903, −7.955501978741571975268752987266, −6.20018864333863322977650211934, −5.92214672161979116844638728930, −4.56086008360934674249613571156, −4.07731768956973996107317184060, −2.82046639027671690133603513918, −1.38291540500776449405042379443,
0.14009834903949320378543697717, 2.16803855450648231687000867541, 3.42917036206740453150767168221, 4.55713085881649760738210604814, 5.51477972287275976326694837783, 6.58212141562674271682023970794, 7.02080809340228534739424181541, 7.61300534963979971685244913232, 8.642761691423529028332524085648, 9.770664330518859348234473610828