L(s) = 1 | + (−0.0966 − 0.297i)2-s + (1.44 + 1.04i)3-s + (0.729 − 0.530i)4-s + (−0.309 + 0.951i)5-s + (0.172 − 0.530i)6-s + (−0.481 − 0.349i)8-s + (0.672 + 2.06i)9-s + 0.312·10-s + (−0.951 + 0.309i)11-s + 1.60·12-s + (−0.610 − 1.87i)13-s + (−1.44 + 1.04i)15-s + (0.221 − 0.680i)16-s + (0.550 − 0.400i)18-s + (0.809 + 0.587i)19-s + (0.278 + 0.857i)20-s + ⋯ |
L(s) = 1 | + (−0.0966 − 0.297i)2-s + (1.44 + 1.04i)3-s + (0.729 − 0.530i)4-s + (−0.309 + 0.951i)5-s + (0.172 − 0.530i)6-s + (−0.481 − 0.349i)8-s + (0.672 + 2.06i)9-s + 0.312·10-s + (−0.951 + 0.309i)11-s + 1.60·12-s + (−0.610 − 1.87i)13-s + (−1.44 + 1.04i)15-s + (0.221 − 0.680i)16-s + (0.550 − 0.400i)18-s + (0.809 + 0.587i)19-s + (0.278 + 0.857i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.601707506\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.601707506\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.951 - 0.309i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.437 + 1.34i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + 0.907T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.20257237821876434742693938324, −9.818780935666514845394682619348, −8.537190168534896605772195296610, −7.70152215779545008142666836817, −7.27550969686240609691261226034, −5.77389864200690230592362813246, −4.91838525998672069329015057856, −3.43545127897419064151869777948, −3.00901781819505191858850658289, −2.20755853854052486369819765413,
1.66268835799979174627563984981, 2.53813625316108690803939248694, 3.51326488773457440377159413013, 4.69990473911418829559275618616, 6.11555261953621001051012350383, 7.18261516633785606741438118937, 7.49328675340264968820673108079, 8.286760070362749722851911535171, 8.984773478167463834883819787619, 9.480697769801395551483617134084