L(s) = 1 | + (0.234 + 0.645i)2-s + (1.17 − 0.982i)4-s + (−0.231 − 1.31i)5-s + (1.38 + 2.25i)7-s + (2.09 + 1.21i)8-s + (0.793 − 0.458i)10-s + (−0.216 − 0.0381i)11-s + (0.673 − 1.85i)13-s + (−1.13 + 1.42i)14-s + (0.242 − 1.37i)16-s + (0.469 + 0.813i)17-s + (3.04 + 1.75i)19-s + (−1.56 − 1.31i)20-s + (−0.0262 − 0.148i)22-s + (−1.17 − 1.40i)23-s + ⋯ |
L(s) = 1 | + (0.166 + 0.456i)2-s + (0.585 − 0.491i)4-s + (−0.103 − 0.588i)5-s + (0.521 + 0.852i)7-s + (0.741 + 0.428i)8-s + (0.251 − 0.144i)10-s + (−0.0652 − 0.0115i)11-s + (0.186 − 0.513i)13-s + (−0.302 + 0.379i)14-s + (0.0605 − 0.343i)16-s + (0.113 + 0.197i)17-s + (0.697 + 0.402i)19-s + (−0.349 − 0.293i)20-s + (−0.00558 − 0.0316i)22-s + (−0.245 − 0.292i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99484 + 0.132754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99484 + 0.132754i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.38 - 2.25i)T \) |
good | 2 | \( 1 + (-0.234 - 0.645i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.231 + 1.31i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (0.216 + 0.0381i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.673 + 1.85i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.469 - 0.813i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.04 - 1.75i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.17 + 1.40i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.33 - 3.66i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.95 + 7.09i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.56 + 2.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.8 - 3.95i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.261 - 1.48i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.18 + 4.35i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 3.77iT - 53T^{2} \) |
| 59 | \( 1 + (-1.72 - 9.76i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.04 - 7.20i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (6.36 + 2.31i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (11.8 - 6.85i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.68 - 4.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.91 - 2.88i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (16.1 - 5.88i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.46 + 5.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.42 + 0.780i)T + (91.1 + 33.1i)T^{2} \) |
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
−10.86145485877351477730112543848, −9.896100567643162663672556855064, −8.861474503118441878017308599365, −8.037911144892603247445092576591, −7.21649857727250909955496221225, −5.90850148327915004309488138449, −5.47840918989907103931920170570, −4.41222003871875480119253718301, −2.72151477799776292920438585126, −1.39529206755886985430225572194,
1.52011896800860769234037208073, 2.91404077091349791514552170120, 3.82416938443749545190594781600, 4.88723117518086760480901073716, 6.40567591702842283218522997733, 7.27172087736784628101622814853, 7.75466357928702880206783439892, 9.025160300638560140142098132202, 10.18874414853396948188953059722, 10.92545954901414206522988113867