| 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 | \( 1 + (4.42 - 0.780i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.92545954901414206522988113867, −10.18874414853396948188953059722, −9.025160300638560140142098132202, −7.75466357928702880206783439892, −7.27172087736784628101622814853, −6.40567591702842283218522997733, −4.88723117518086760480901073716, −3.82416938443749545190594781600, −2.91404077091349791514552170120, −1.52011896800860769234037208073,
1.39529206755886985430225572194, 2.72151477799776292920438585126, 4.41222003871875480119253718301, 5.47840918989907103931920170570, 5.90850148327915004309488138449, 7.21649857727250909955496221225, 8.037911144892603247445092576591, 8.861474503118441878017308599365, 9.896100567643162663672556855064, 10.86145485877351477730112543848
