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
−10.54663222580306048055134419710, −10.02801096958636131013272000831, −9.521487542188233920553263081802, −7.938348927425553897573909593706, −7.12361287653074844043074653639, −6.23244671441294247383288114387, −5.16168448942085024887371820339, −3.35971577100309043524689253930, −2.21727577023369613238101656973, −0.808250397132473998634649886204,
0.937701419882341279135477638063, 3.24680232074705900433647398860, 5.14089930833169014017195613128, 5.75477434223663301283514047509, 6.34099837778122629448953090478, 7.52675835069519305041710103954, 8.621992467229916376774511515472, 9.377451333626533810583329227934, 9.855805476249862688426890413802, 10.73228194014305922544313535526