L(s) = 1 | + (−1.42 + 1.42i)2-s + (−1.57 + 0.730i)3-s − 2.07i·4-s + (1.72 − 1.72i)5-s + (1.19 − 3.28i)6-s + (−2.20 + 2.20i)7-s + (0.105 + 0.105i)8-s + (1.93 − 2.29i)9-s + 4.91i·10-s + (−1.95 − 1.95i)11-s + (1.51 + 3.25i)12-s − 6.28i·14-s + (−1.44 + 3.96i)15-s + 3.84·16-s + 5.78·17-s + (0.515 + 6.03i)18-s + ⋯ |
L(s) = 1 | + (−1.00 + 1.00i)2-s + (−0.906 + 0.421i)3-s − 1.03i·4-s + (0.770 − 0.770i)5-s + (0.489 − 1.34i)6-s + (−0.831 + 0.831i)7-s + (0.0372 + 0.0372i)8-s + (0.644 − 0.764i)9-s + 1.55i·10-s + (−0.590 − 0.590i)11-s + (0.437 + 0.940i)12-s − 1.67i·14-s + (−0.373 + 1.02i)15-s + 0.961·16-s + 1.40·17-s + (0.121 + 1.42i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.438568 + 0.419226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.438568 + 0.419226i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.57 - 0.730i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.42 - 1.42i)T - 2iT^{2} \) |
| 5 | \( 1 + (-1.72 + 1.72i)T - 5iT^{2} \) |
| 7 | \( 1 + (2.20 - 2.20i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.95 + 1.95i)T + 11iT^{2} \) |
| 17 | \( 1 - 5.78T + 17T^{2} \) |
| 19 | \( 1 + (-1.06 - 1.06i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.86T + 23T^{2} \) |
| 29 | \( 1 + 2.92iT - 29T^{2} \) |
| 31 | \( 1 + (-3.56 - 3.56i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.83 - 2.83i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.79 + 4.79i)T - 41iT^{2} \) |
| 43 | \( 1 - 1.84iT - 43T^{2} \) |
| 47 | \( 1 + (-0.115 - 0.115i)T + 47iT^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 + (1.22 + 1.22i)T + 59iT^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + (-9.65 - 9.65i)T + 67iT^{2} \) |
| 71 | \( 1 + (0.239 - 0.239i)T - 71iT^{2} \) |
| 73 | \( 1 + (-8.54 + 8.54i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + (-2.83 + 2.83i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.00 - 3.00i)T + 89iT^{2} \) |
| 97 | \( 1 + (-6.99 - 6.99i)T + 97iT^{2} \) |
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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
−10.73228194014305922544313535526, −9.855805476249862688426890413802, −9.377451333626533810583329227934, −8.621992467229916376774511515472, −7.52675835069519305041710103954, −6.34099837778122629448953090478, −5.75477434223663301283514047509, −5.14089930833169014017195613128, −3.24680232074705900433647398860, −0.937701419882341279135477638063,
0.808250397132473998634649886204, 2.21727577023369613238101656973, 3.35971577100309043524689253930, 5.16168448942085024887371820339, 6.23244671441294247383288114387, 7.12361287653074844043074653639, 7.938348927425553897573909593706, 9.521487542188233920553263081802, 10.02801096958636131013272000831, 10.54663222580306048055134419710