L(s) = 1 | + (−1.01 + 0.985i)2-s + 3-s + (0.0573 − 1.99i)4-s − 1.66i·5-s + (−1.01 + 0.985i)6-s + 0.193·7-s + (1.91 + 2.08i)8-s + 9-s + (1.64 + 1.69i)10-s + 5.43·11-s + (0.0573 − 1.99i)12-s + 3.51i·13-s + (−0.196 + 0.190i)14-s − 1.66i·15-s + (−3.99 − 0.229i)16-s − 4.63·17-s + ⋯ |
L(s) = 1 | + (−0.717 + 0.696i)2-s + 0.577·3-s + (0.0286 − 0.999i)4-s − 0.746i·5-s + (−0.414 + 0.402i)6-s + 0.0731·7-s + (0.676 + 0.736i)8-s + 0.333·9-s + (0.520 + 0.535i)10-s + 1.63·11-s + (0.0165 − 0.577i)12-s + 0.975i·13-s + (−0.0524 + 0.0510i)14-s − 0.430i·15-s + (−0.998 − 0.0573i)16-s − 1.12·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43210 + 0.0487418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43210 + 0.0487418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.01 - 0.985i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (-0.790 - 8.14i)T \) |
good | 5 | \( 1 + 1.66iT - 5T^{2} \) |
| 7 | \( 1 - 0.193T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 - 3.51iT - 13T^{2} \) |
| 17 | \( 1 + 4.63T + 17T^{2} \) |
| 19 | \( 1 + 1.14iT - 19T^{2} \) |
| 23 | \( 1 + 7.82iT - 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 0.746iT - 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 - 2.28iT - 47T^{2} \) |
| 53 | \( 1 + 6.58iT - 53T^{2} \) |
| 59 | \( 1 + 1.92iT - 59T^{2} \) |
| 61 | \( 1 + 1.81iT - 61T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 - 5.05T + 73T^{2} \) |
| 79 | \( 1 + 8.43T + 79T^{2} \) |
| 83 | \( 1 - 8.12iT - 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 - 8.73iT - 97T^{2} \) |
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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
−9.824300732640384237602356267352, −9.104211277347421304275664453355, −8.777582404644151280470088636624, −7.911882049546242872903280230185, −6.62544753394453605440559898070, −6.42947890215594834075144233208, −4.72374130944765833560894411190, −4.23810417390396754638382265688, −2.27395195931714349247755699363, −1.04778954713904512512753500802,
1.30603122258834577026268375626, 2.60792737120885997419027619552, 3.50503100279207821468107597393, 4.38177184180353388478021461382, 6.16012679675683359111137628724, 7.05485162704353878815776201131, 7.82235589074452353972522249650, 8.735909831090441011677038396433, 9.463465681503527456180599674774, 10.10155965043259261585594641757