L(s) = 1 | − 2-s − 1.98·3-s + 4-s − 2.70·5-s + 1.98·6-s + 2.97·7-s − 8-s + 0.924·9-s + 2.70·10-s − 1.51·11-s − 1.98·12-s + 5.78·13-s − 2.97·14-s + 5.34·15-s + 16-s − 3.10·17-s − 0.924·18-s + 19-s − 2.70·20-s − 5.88·21-s + 1.51·22-s − 4.86·23-s + 1.98·24-s + 2.29·25-s − 5.78·26-s + 4.11·27-s + 2.97·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.20·5-s + 0.808·6-s + 1.12·7-s − 0.353·8-s + 0.308·9-s + 0.853·10-s − 0.457·11-s − 0.571·12-s + 1.60·13-s − 0.793·14-s + 1.38·15-s + 0.250·16-s − 0.754·17-s − 0.217·18-s + 0.229·19-s − 0.603·20-s − 1.28·21-s + 0.323·22-s − 1.01·23-s + 0.404·24-s + 0.458·25-s − 1.13·26-s + 0.791·27-s + 0.561·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6365854659\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6365854659\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 - 5.78T + 13T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 0.914T + 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 5.49T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 1.66T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 + 5.64T + 97T^{2} \) |
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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
−7.954542342987462290843299119354, −7.33466115235161149526741125259, −6.33394490667937315725228629476, −5.96030061987859328420466453269, −5.06816861610080579450043726912, −4.32876244023697190732108886342, −3.69731141837021505408223380973, −2.49316618767106413809850410941, −1.36366797531010831778702018407, −0.51266045314011340134580379853,
0.51266045314011340134580379853, 1.36366797531010831778702018407, 2.49316618767106413809850410941, 3.69731141837021505408223380973, 4.32876244023697190732108886342, 5.06816861610080579450043726912, 5.96030061987859328420466453269, 6.33394490667937315725228629476, 7.33466115235161149526741125259, 7.954542342987462290843299119354