L(s) = 1 | − 26·3-s − 22·7-s + 433·9-s + 768·11-s + 46·13-s − 378·17-s − 1.10e3·19-s + 572·21-s − 1.98e3·23-s − 4.94e3·27-s − 5.61e3·29-s + 3.98e3·31-s − 1.99e4·33-s + 142·37-s − 1.19e3·39-s + 1.54e3·41-s − 5.02e3·43-s + 2.47e4·47-s − 1.63e4·49-s + 9.82e3·51-s + 1.41e4·53-s + 2.86e4·57-s − 2.83e4·59-s + 5.52e3·61-s − 9.52e3·63-s − 2.47e4·67-s + 5.16e4·69-s + ⋯ |
L(s) = 1 | − 1.66·3-s − 0.169·7-s + 1.78·9-s + 1.91·11-s + 0.0754·13-s − 0.317·17-s − 0.699·19-s + 0.283·21-s − 0.782·23-s − 1.30·27-s − 1.23·29-s + 0.745·31-s − 3.19·33-s + 0.0170·37-s − 0.125·39-s + 0.143·41-s − 0.414·43-s + 1.63·47-s − 0.971·49-s + 0.529·51-s + 0.692·53-s + 1.16·57-s − 1.06·59-s + 0.190·61-s − 0.302·63-s − 0.673·67-s + 1.30·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.012626024\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.012626024\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 26 T + p^{5} T^{2} \) |
| 7 | \( 1 + 22 T + p^{5} T^{2} \) |
| 11 | \( 1 - 768 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 + 378 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1100 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1986 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5610 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3988 T + p^{5} T^{2} \) |
| 37 | \( 1 - 142 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1542 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5026 T + p^{5} T^{2} \) |
| 47 | \( 1 - 24738 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14166 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5522 T + p^{5} T^{2} \) |
| 67 | \( 1 + 24742 T + p^{5} T^{2} \) |
| 71 | \( 1 + 42372 T + p^{5} T^{2} \) |
| 73 | \( 1 - 52126 T + p^{5} T^{2} \) |
| 79 | \( 1 - 39640 T + p^{5} T^{2} \) |
| 83 | \( 1 + 59826 T + p^{5} T^{2} \) |
| 89 | \( 1 - 57690 T + p^{5} T^{2} \) |
| 97 | \( 1 - 144382 T + p^{5} T^{2} \) |
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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
−10.66049697553484844364589976630, −9.722829854459598029461233029726, −8.797512192057952053313099815564, −7.30103624781289634156379294086, −6.38477507214835223463864881094, −5.94135384705747554999519759253, −4.63348310676988706331941361674, −3.82671477962914164662221660740, −1.73680774699113484770365600003, −0.59847245558727495978595998991,
0.59847245558727495978595998991, 1.73680774699113484770365600003, 3.82671477962914164662221660740, 4.63348310676988706331941361674, 5.94135384705747554999519759253, 6.38477507214835223463864881094, 7.30103624781289634156379294086, 8.797512192057952053313099815564, 9.722829854459598029461233029726, 10.66049697553484844364589976630