L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.396 − 1.22i)3-s + (0.309 − 0.951i)4-s + (1.09 + 0.799i)5-s + (−1.03 − 0.754i)6-s + (0.979 − 3.01i)7-s + (−0.309 − 0.951i)8-s + (1.09 − 0.794i)9-s + 1.35·10-s + (−0.286 + 3.30i)11-s − 1.28·12-s + (−0.734 + 0.533i)13-s + (−0.979 − 3.01i)14-s + (0.539 − 1.65i)15-s + (−0.809 − 0.587i)16-s + (−2.29 − 1.66i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.229 − 0.705i)3-s + (0.154 − 0.475i)4-s + (0.491 + 0.357i)5-s + (−0.424 − 0.308i)6-s + (0.370 − 1.13i)7-s + (−0.109 − 0.336i)8-s + (0.364 − 0.264i)9-s + 0.429·10-s + (−0.0863 + 0.996i)11-s − 0.370·12-s + (−0.203 + 0.147i)13-s + (−0.261 − 0.805i)14-s + (0.139 − 0.428i)15-s + (−0.202 − 0.146i)16-s + (−0.556 − 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29190 - 1.44275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29190 - 1.44275i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.286 - 3.30i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (0.396 + 1.22i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.09 - 0.799i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.979 + 3.01i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.734 - 0.533i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.29 + 1.66i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + (1.53 - 4.72i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.34 + 4.60i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.93 - 5.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.33 - 10.2i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.172T + 43T^{2} \) |
| 47 | \( 1 + (1.78 + 5.48i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (10.9 - 7.93i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.248 + 0.765i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.09 - 5.88i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + (-10.6 - 7.75i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.09 - 3.37i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (4.58 - 3.33i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.58 - 1.88i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 + (3.27 - 2.37i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.09525991020092172814009266184, −10.14967809678910074403521644473, −9.549347529489839139185724612134, −7.895328164032726753691692058487, −6.91902420452125110446908743623, −6.49180571126327360292384361837, −4.94147399960320750186973654018, −4.11542436103591130074043960107, −2.49581010963308470965622130496, −1.22407830019876411678379490936,
2.16188007768039675136075842647, 3.65496815426679802872303010740, 4.94774578146311222533026123721, 5.47691310254806696547500342193, 6.40903214757587815523999559872, 7.83792437010697106597707979427, 8.750004905807792204651042265242, 9.509509139579234930793000775268, 10.70347539268771819292339501412, 11.39963871612799372688490201496