L(s) = 1 | + (−0.0537 + 1.73i)3-s + (−0.560 − 0.560i)7-s + (−2.99 − 0.186i)9-s + 0.939i·11-s + (−4.13 + 4.13i)13-s + (−4.50 + 4.50i)17-s − 3.74i·19-s + (0.999 − 0.939i)21-s + (6.18 + 6.18i)23-s + (0.483 − 5.17i)27-s − 3.16·29-s − 5.37·31-s + (−1.62 − 0.0505i)33-s + (−3.56 − 3.56i)37-s + (−6.92 − 7.37i)39-s + ⋯ |
L(s) = 1 | + (−0.0310 + 0.999i)3-s + (−0.211 − 0.211i)7-s + (−0.998 − 0.0620i)9-s + 0.283i·11-s + (−1.14 + 1.14i)13-s + (−1.09 + 1.09i)17-s − 0.859i·19-s + (0.218 − 0.205i)21-s + (1.28 + 1.28i)23-s + (0.0929 − 0.995i)27-s − 0.588·29-s − 0.964·31-s + (−0.283 − 0.00879i)33-s + (−0.586 − 0.586i)37-s + (−1.10 − 1.18i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.142309 + 0.774253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.142309 + 0.774253i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0537 - 1.73i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.560 + 0.560i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.939iT - 11T^{2} \) |
| 13 | \( 1 + (4.13 - 4.13i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.50 - 4.50i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.74iT - 19T^{2} \) |
| 23 | \( 1 + (-6.18 - 6.18i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 5.37T + 31T^{2} \) |
| 37 | \( 1 + (3.56 + 3.56i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.15iT - 41T^{2} \) |
| 43 | \( 1 + (-4.33 + 4.33i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.65 - 5.65i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.526 + 0.526i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 + (-1.22 - 1.22i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.16iT - 71T^{2} \) |
| 73 | \( 1 + (5.56 - 5.56i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.74iT - 79T^{2} \) |
| 83 | \( 1 + (3.97 + 3.97i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + (-5.25 - 5.25i)T + 97iT^{2} \) |
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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
−11.12671243206900673422676694760, −10.08438583868596019777047374677, −9.310832939129963960153579646354, −8.807077880341486041265916974683, −7.40731546948309400142020246622, −6.60507081698553352898278400806, −5.31986518357305094508193734857, −4.51435934849797575708244934514, −3.57712474469355409095219285154, −2.18609561441516681576576206721,
0.41095841807817352269418633608, 2.24103090954395985859309027814, 3.16368262675598519071201522650, 4.90343513477726460988932894588, 5.74480250648468594093805205220, 6.85808192142902942414773778089, 7.45809293107743920382031470300, 8.479810259890791087075201720949, 9.232190646718315358147179689211, 10.39765038579265042663697147507