| L(s) = 1 | + (−0.903 + 2.78i)2-s + (−3.67 + 2.66i)3-s + (−0.441 − 0.320i)4-s + (1.90 + 5.87i)5-s + (−4.10 − 12.6i)6-s + (25.9 + 18.8i)7-s + (−17.6 + 12.8i)8-s + (−1.97 + 6.06i)9-s − 18.0·10-s + 2.47·12-s + (1.38 − 4.27i)13-s + (−75.9 + 55.2i)14-s + (−22.6 − 16.4i)15-s + (−21.0 − 64.7i)16-s + (−18.3 − 56.5i)17-s + (−15.0 − 10.9i)18-s + ⋯ |
| L(s) = 1 | + (−0.319 + 0.982i)2-s + (−0.707 + 0.513i)3-s + (−0.0552 − 0.0401i)4-s + (0.170 + 0.525i)5-s + (−0.279 − 0.859i)6-s + (1.40 + 1.01i)7-s + (−0.779 + 0.566i)8-s + (−0.0729 + 0.224i)9-s − 0.571·10-s + 0.0596·12-s + (0.0296 − 0.0912i)13-s + (−1.45 + 1.05i)14-s + (−0.390 − 0.283i)15-s + (−0.328 − 1.01i)16-s + (−0.262 − 0.807i)17-s + (−0.197 − 0.143i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.168135 - 1.14272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.168135 - 1.14272i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (0.903 - 2.78i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (3.67 - 2.66i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (-1.90 - 5.87i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-25.9 - 18.8i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (-1.38 + 4.27i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (18.3 + 56.5i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-23.4 + 17.0i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 38.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-31.9 - 23.1i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (82.2 - 253. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (90.7 + 65.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (108. - 78.8i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-147. + 107. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-13.2 + 40.6i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-146. - 106. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-172. - 532. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 770.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (8.20 + 25.2i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-301. - 218. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-78.1 + 240. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (326. + 1.00e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 58.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + (184. - 568. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−14.05837542918084044404395639627, −12.13127815426655003314786478914, −11.42730938848820118698623380661, −10.58230768536018994869155351318, −9.009443199129599355546432451501, −8.116537939725918622401098889655, −6.94795154558378435933318752305, −5.64687258657317382990416091397, −4.95419390262777762594646314504, −2.48647625507708879729371783592,
0.76069222663451007427017088179, 1.75352082814126564457655374081, 3.99587620510653349664298913230, 5.53688045664167290906369243441, 6.84905445669867924231140472181, 8.173450808537757203148107631352, 9.481701510767839586935313431810, 10.74331582326830743560863403536, 11.27169772701067412191124993929, 12.16980230279183141882420344140
