L(s) = 1 | + (0.456 + 1.33i)2-s + (−0.707 − 0.707i)3-s + (−1.58 + 1.22i)4-s + (1.65 − 1.50i)5-s + (0.623 − 1.26i)6-s + 2.58·7-s + (−2.35 − 1.55i)8-s + 1.00i·9-s + (2.76 + 1.53i)10-s + (4.39 + 4.39i)11-s + (1.98 + 0.254i)12-s + (0.417 + 0.417i)13-s + (1.18 + 3.46i)14-s + (−2.23 − 0.110i)15-s + (1.00 − 3.87i)16-s − 4.40i·17-s + ⋯ |
L(s) = 1 | + (0.323 + 0.946i)2-s + (−0.408 − 0.408i)3-s + (−0.791 + 0.611i)4-s + (0.741 − 0.671i)5-s + (0.254 − 0.518i)6-s + 0.978·7-s + (−0.834 − 0.551i)8-s + 0.333i·9-s + (0.874 + 0.484i)10-s + (1.32 + 1.32i)11-s + (0.572 + 0.0734i)12-s + (0.115 + 0.115i)13-s + (0.316 + 0.926i)14-s + (−0.576 − 0.0286i)15-s + (0.252 − 0.967i)16-s − 1.06i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31791 + 0.615234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31791 + 0.615234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.456 - 1.33i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.65 + 1.50i)T \) |
good | 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 + (-4.39 - 4.39i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.417 - 0.417i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.40iT - 17T^{2} \) |
| 19 | \( 1 + (4.53 - 4.53i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 + (-3.73 + 3.73i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + (5.26 - 5.26i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.16iT - 41T^{2} \) |
| 43 | \( 1 + (2.66 - 2.66i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.45iT - 47T^{2} \) |
| 53 | \( 1 + (2.89 - 2.89i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.60 + 4.60i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.211 - 0.211i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.17 + 7.17i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 + (4.56 + 4.56i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 6.78iT - 97T^{2} \) |
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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
−12.27073962227662985436977454576, −11.81245760433474292264619303888, −10.11998303382172906792178205048, −9.131165840007377090147210201494, −8.223369151266514383190173178355, −7.08368858609646375468937797917, −6.21474690329066604446733738917, −5.05214612703059771120254200805, −4.30744080537319655180035187289, −1.71438705236091204912060817954,
1.57845476049274832266405837581, 3.26215502631623561500228519336, 4.46131006306247884532694875343, 5.71606955953733953093652752652, 6.52036187673477654342208087112, 8.545616129177440884995955333839, 9.220233609711683303320128178013, 10.59362878013374763064707020657, 10.92259079693704072243046512847, 11.69245035650146823636866448403