L(s) = 1 | + (0.809 + 0.587i)2-s + (1.02 − 3.14i)3-s + (0.309 + 0.951i)4-s + (0.670 − 0.487i)5-s + (2.67 − 1.94i)6-s + (−0.688 − 2.11i)7-s + (−0.309 + 0.951i)8-s + (−6.42 − 4.66i)9-s + 0.828·10-s + (0.689 + 3.24i)11-s + 3.30·12-s + (0.976 + 0.709i)13-s + (0.688 − 2.11i)14-s + (−0.847 − 2.60i)15-s + (−0.809 + 0.587i)16-s + (3.70 − 2.69i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.590 − 1.81i)3-s + (0.154 + 0.475i)4-s + (0.299 − 0.217i)5-s + (1.09 − 0.793i)6-s + (−0.260 − 0.801i)7-s + (−0.109 + 0.336i)8-s + (−2.14 − 1.55i)9-s + 0.262·10-s + (0.207 + 0.978i)11-s + 0.954·12-s + (0.270 + 0.196i)13-s + (0.184 − 0.566i)14-s + (−0.218 − 0.673i)15-s + (−0.202 + 0.146i)16-s + (0.899 − 0.653i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73984 - 1.37506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73984 - 1.37506i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-0.689 - 3.24i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-1.02 + 3.14i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.670 + 0.487i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.688 + 2.11i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.976 - 0.709i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.70 + 2.69i)T + (5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + (1.20 + 3.72i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-7.02 - 5.10i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.423 + 1.30i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.82 + 5.62i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 + (2.44 - 7.53i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.58 - 4.78i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.0323 + 0.0994i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-10.1 + 7.33i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + (4.36 - 3.17i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.09 - 12.6i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.87 + 1.36i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.84 - 3.51i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 6.07T + 89T^{2} \) |
| 97 | \( 1 + (14.0 + 10.1i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.50080896568557776058377450456, −9.945867868579326148677957582401, −8.901185325284198235621940219233, −7.81585220369433311060510571432, −7.26313942023310575963413345002, −6.54023206528354212273800630628, −5.53644926527062409125183623375, −3.92636224230624288318406605364, −2.62230777685407939139633175225, −1.28803283484555264954634437764,
2.57371333283428744307128441767, 3.38936789129391791949998502194, 4.31647212587368318153838993420, 5.53343431960702723780379330775, 6.07499712676102310725425058667, 8.198300680599022066314964162368, 8.867703773890097271624034296773, 9.851676059475983835676278971078, 10.34611436723617489696467109271, 11.24705916459290239721021146581