| L(s) = 1 | + (0.0599 − 1.41i)2-s + (1.04 + 1.38i)3-s + (−1.99 − 0.169i)4-s + (−2.46 + 0.324i)5-s + (2.01 − 1.39i)6-s + (−4.86 + 1.30i)7-s + (−0.359 + 2.80i)8-s + (−0.824 + 2.88i)9-s + (0.310 + 3.49i)10-s + (1.56 − 1.20i)11-s + (−1.84 − 2.93i)12-s + (−1.11 + 1.44i)13-s + (1.55 + 6.95i)14-s + (−3.01 − 3.06i)15-s + (3.94 + 0.675i)16-s + 1.47i·17-s + ⋯ |
| L(s) = 1 | + (0.0424 − 0.999i)2-s + (0.602 + 0.798i)3-s + (−0.996 − 0.0847i)4-s + (−1.10 + 0.144i)5-s + (0.823 − 0.567i)6-s + (−1.84 + 0.493i)7-s + (−0.126 + 0.991i)8-s + (−0.274 + 0.961i)9-s + (0.0980 + 1.10i)10-s + (0.471 − 0.361i)11-s + (−0.532 − 0.846i)12-s + (−0.308 + 0.402i)13-s + (0.414 + 1.85i)14-s + (−0.778 − 0.791i)15-s + (0.985 + 0.168i)16-s + 0.358i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.267979 + 0.369835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267979 + 0.369835i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0599 + 1.41i)T \) |
| 3 | \( 1 + (-1.04 - 1.38i)T \) |
| good | 5 | \( 1 + (2.46 - 0.324i)T + (4.82 - 1.29i)T^{2} \) |
| 7 | \( 1 + (4.86 - 1.30i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 1.20i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.11 - 1.44i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 1.47iT - 17T^{2} \) |
| 19 | \( 1 + (0.944 + 2.27i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 0.575i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.411 - 3.12i)T + (-28.0 - 7.50i)T^{2} \) |
| 31 | \( 1 + (4.42 + 7.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.68 - 4.07i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-4.34 - 1.16i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (7.57 - 5.81i)T + (11.1 - 41.5i)T^{2} \) |
| 47 | \( 1 + (-11.4 - 6.62i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.35 - 3.04i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (9.56 - 1.25i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (0.966 - 7.33i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (1.84 + 1.41i)T + (17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (8.91 + 8.91i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.38 - 3.38i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.870 + 0.502i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (13.0 + 1.72i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-5.73 - 5.73i)T + 89iT^{2} \) |
| 97 | \( 1 + (-2.21 + 3.82i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.97816971664138314665614765601, −11.15059330839447059418238809826, −10.18870314924671248706570297246, −9.263364771463577473576797188749, −8.837530913546889642964388627116, −7.48058478107607221599051577608, −5.93921055162269675249779158273, −4.36615418632407822075500153941, −3.54143986073111513966815141313, −2.74447282660214488798608463585,
0.30729034368438421642635936700, 3.28811752089146235712749274799, 4.04303721327489232013867637172, 5.85954362219144895949141845996, 7.09802985078028261967012885172, 7.20972948055899466867292184649, 8.502141107508911557598907476893, 9.289985790097148675040845730214, 10.25834193279194179104191033430, 12.12113599185975508590659550897
