L(s) = 1 | + (1.27 − 1.27i)2-s − 1.24i·4-s + (−2.19 − 0.428i)5-s + (0.936 + 0.936i)7-s + (0.958 + 0.958i)8-s + (−3.34 + 2.25i)10-s + 0.533i·11-s + (4.37 − 4.37i)13-s + 2.38·14-s + 4.93·16-s + (−0.921 + 0.921i)17-s − 7.99i·19-s + (−0.534 + 2.73i)20-s + (0.679 + 0.679i)22-s + (−0.707 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.901 − 0.901i)2-s − 0.623i·4-s + (−0.981 − 0.191i)5-s + (0.354 + 0.354i)7-s + (0.339 + 0.339i)8-s + (−1.05 + 0.711i)10-s + 0.160i·11-s + (1.21 − 1.21i)13-s + 0.637·14-s + 1.23·16-s + (−0.223 + 0.223i)17-s − 1.83i·19-s + (−0.119 + 0.612i)20-s + (0.144 + 0.144i)22-s + (−0.147 − 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.469500508\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.469500508\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.19 + 0.428i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1.27 + 1.27i)T - 2iT^{2} \) |
| 7 | \( 1 + (-0.936 - 0.936i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.533iT - 11T^{2} \) |
| 13 | \( 1 + (-4.37 + 4.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.921 - 0.921i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.99iT - 19T^{2} \) |
| 29 | \( 1 - 9.15T + 29T^{2} \) |
| 31 | \( 1 - 2.22T + 31T^{2} \) |
| 37 | \( 1 + (2.15 + 2.15i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.2iT - 41T^{2} \) |
| 43 | \( 1 + (-5.40 + 5.40i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.53 - 1.53i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.98 + 2.98i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.39T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + (-9.65 - 9.65i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.85iT - 71T^{2} \) |
| 73 | \( 1 + (9.68 - 9.68i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.38iT - 79T^{2} \) |
| 83 | \( 1 + (8.08 + 8.08i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + (-1.22 - 1.22i)T + 97iT^{2} \) |
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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
−10.11004893699339798370534466794, −8.647766854641080615068456713058, −8.320685290632309574743253829348, −7.30011860446968091906371836397, −6.11029509870228791759356291762, −4.96524267002237793696764621649, −4.44320729375817894013132316987, −3.35965710530192380034135476970, −2.64505898392295288224115457976, −1.02751205833221636324772705748,
1.37291694204688428369763437133, 3.38334255748426736009570413735, 4.13780427870046891224718961776, 4.74085046397362445676447809028, 6.05500344188437846783128950866, 6.53223335223924794890605261245, 7.53759208262587548827129448881, 8.098378946541692941024780966782, 9.025575493543640029140424457037, 10.34674651156502976261335192703