| L(s) = 1 | + (0.866 − 1.23i)2-s + (−0.938 + 1.45i)3-s + (−0.0960 − 0.264i)4-s + (0.525 + 2.17i)5-s + (0.987 + 2.42i)6-s + (1.45 − 0.677i)7-s + (2.50 + 0.671i)8-s + (−1.23 − 2.73i)9-s + (3.14 + 1.23i)10-s + (−1.78 − 2.12i)11-s + (0.474 + 0.107i)12-s + (−0.251 + 0.176i)13-s + (0.420 − 2.38i)14-s + (−3.65 − 1.27i)15-s + (3.43 − 2.88i)16-s + (−7.02 + 1.88i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.874i)2-s + (−0.542 + 0.840i)3-s + (−0.0480 − 0.132i)4-s + (0.234 + 0.972i)5-s + (0.403 + 0.988i)6-s + (0.549 − 0.256i)7-s + (0.886 + 0.237i)8-s + (−0.412 − 0.911i)9-s + (0.994 + 0.389i)10-s + (−0.538 − 0.641i)11-s + (0.136 + 0.0311i)12-s + (−0.0698 + 0.0489i)13-s + (0.112 − 0.637i)14-s + (−0.944 − 0.329i)15-s + (0.858 − 0.720i)16-s + (−1.70 + 0.456i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35948 + 0.0800350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35948 + 0.0800350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.938 - 1.45i)T \) |
| 5 | \( 1 + (-0.525 - 2.17i)T \) |
| good | 2 | \( 1 + (-0.866 + 1.23i)T + (-0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (-1.45 + 0.677i)T + (4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (1.78 + 2.12i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.251 - 0.176i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (7.02 - 1.88i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.07 + 2.92i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0543 - 0.116i)T + (-14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (1.51 + 8.60i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.22 + 1.53i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (0.558 + 2.08i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (4.62 + 0.815i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.90 + 0.429i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.58 - 7.69i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.483 - 0.483i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.43 + 6.23i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.50 - 0.910i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.619 - 0.884i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-9.03 - 5.21i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.78 + 6.65i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.04 - 0.360i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 7.55i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-5.58 - 9.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.03 - 11.8i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−13.36212933988821186600403747282, −11.85183840649238169614262683026, −11.11776736461759658884866979107, −10.71866808350471942648973493361, −9.554732806842654245523248488285, −7.921443852304875904542609848549, −6.45779183488923082489727871615, −5.03953337491600169676551299339, −3.89752126833007991894966070357, −2.61240918505480662640426131119,
1.73002228030569204816054887765, 4.83361158247525951421014123939, 5.28831325483019165789867674934, 6.58702128703147186631651748201, 7.56583697254314316926172329126, 8.637901424896336392260460883436, 10.20911879917979717284115008400, 11.48334050194217136064324912506, 12.48066422100741208969125303169, 13.35442898402438766545095653032
