| L(s) = 1 | + (−0.819 − 0.573i)2-s + (1.32 − 1.11i)3-s + (0.342 + 0.939i)4-s + (−1.31 + 1.80i)5-s + (−1.72 + 0.151i)6-s + (1.73 + 3.71i)7-s + (0.258 − 0.965i)8-s + (0.519 − 2.95i)9-s + (2.11 − 0.724i)10-s + (2.47 + 2.95i)11-s + (1.50 + 0.865i)12-s + (1.86 + 2.66i)13-s + (0.711 − 4.03i)14-s + (0.265 + 3.86i)15-s + (−0.766 + 0.642i)16-s + (−1.32 − 4.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.405i)2-s + (0.765 − 0.642i)3-s + (0.171 + 0.469i)4-s + (−0.588 + 0.808i)5-s + (−0.704 + 0.0618i)6-s + (0.654 + 1.40i)7-s + (0.0915 − 0.341i)8-s + (0.173 − 0.984i)9-s + (0.668 − 0.229i)10-s + (0.747 + 0.890i)11-s + (0.433 + 0.249i)12-s + (0.517 + 0.739i)13-s + (0.190 − 1.07i)14-s + (0.0685 + 0.997i)15-s + (−0.191 + 0.160i)16-s + (−0.322 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21652 + 0.0426030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21652 + 0.0426030i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (-1.32 + 1.11i)T \) |
| 5 | \( 1 + (1.31 - 1.80i)T \) |
| good | 7 | \( 1 + (-1.73 - 3.71i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-2.47 - 2.95i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.86 - 2.66i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.32 + 4.95i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.11 + 0.641i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.78 + 2.23i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.692 - 3.92i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.74 + 2.09i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.87 - 1.03i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-8.38 - 1.47i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.0646 + 0.738i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.53 - 0.716i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (2.43 + 2.43i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.25 + 4.40i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-5.18 - 1.88i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.66 + 6.76i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (5.52 + 3.19i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (16.0 + 4.30i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (8.47 - 1.49i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.772 - 1.10i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-6.63 - 11.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.3 + 1.25i)T + (95.5 + 16.8i)T^{2} \) |
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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
−11.86426951366860602930377169861, −11.30543582911500763454799441471, −9.776005008119400733614293360531, −8.985144629374032172532396821153, −8.201621093988796480139778292941, −7.20981927825899775033561903257, −6.35588007860907174730976287246, −4.35044802578661626173948386935, −2.88458657557710433091948070945, −1.90867441538549298255582819067,
1.24067709873733293235072911050, 3.69458206733477174136249981316, 4.40806814104284903106194440990, 5.87564673980677331457532701214, 7.48572775931016238458787638905, 8.205696144775221280886258685996, 8.727372715478598855632997931371, 9.979702179476289647084350366250, 10.73667384956258812813069692595, 11.58402263823415216903765802309
