L(s) = 1 | + 2.56i·3-s − 3.59·9-s − 1.56·11-s − 5.56i·13-s + 7.16i·17-s + 3.16·19-s + 5.73i·23-s − 1.53i·27-s − 1.96·29-s − 0.969·31-s − 4.03i·33-s + 6.70i·37-s + 14.3·39-s + 8.87·41-s − 4.59i·43-s + ⋯ |
L(s) = 1 | + 1.48i·3-s − 1.19·9-s − 0.473·11-s − 1.54i·13-s + 1.73i·17-s + 0.726·19-s + 1.19i·23-s − 0.296i·27-s − 0.365·29-s − 0.174·31-s − 0.701i·33-s + 1.10i·37-s + 2.29·39-s + 1.38·41-s − 0.701i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9150596636\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9150596636\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.56iT - 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 5.56iT - 13T^{2} \) |
| 17 | \( 1 - 7.16iT - 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 5.73iT - 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 + 0.969T + 31T^{2} \) |
| 37 | \( 1 - 6.70iT - 37T^{2} \) |
| 41 | \( 1 - 8.87T + 41T^{2} \) |
| 43 | \( 1 + 4.59iT - 43T^{2} \) |
| 47 | \( 1 - 0.401iT - 47T^{2} \) |
| 53 | \( 1 - 9.53iT - 53T^{2} \) |
| 59 | \( 1 + 4.56T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 0.862iT - 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 - 0.233iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.879937773987448214540852900665, −7.84788919374237491964397497500, −7.62838882090210976160927425449, −6.14279631841226771664282751762, −5.64841070230459418654915989872, −5.01402846487645541758493749042, −4.17064262775714533885744628111, −3.43880066276524434500235494560, −2.87835861540896966195213799200, −1.41046770498297896636218629033,
0.25108002624822005646855123051, 1.31644423013940276716624265067, 2.28815587614601193031911876857, 2.85406855935742010530346649893, 4.17044657230207210331565710610, 4.97173662176779380863803480056, 5.86330268995660947749282085915, 6.61844428076656750814054566549, 7.17331775187993384222643078682, 7.60719359716237947480977937656