L(s) = 1 | + (0.707 + 0.707i)2-s + (0.325 − 0.325i)3-s + 1.00i·4-s + (−2.06 − 0.863i)5-s + 0.459·6-s + (2.30 + 2.30i)7-s + (−0.707 + 0.707i)8-s + 2.78i·9-s + (−0.847 − 2.06i)10-s + 5.14·11-s + (0.325 + 0.325i)12-s + (−1.01 − 1.01i)13-s + 3.26i·14-s + (−0.951 + 0.389i)15-s − 1.00·16-s + (−1.10 + 1.10i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.187 − 0.187i)3-s + 0.500i·4-s + (−0.922 − 0.386i)5-s + 0.187·6-s + (0.872 + 0.872i)7-s + (−0.250 + 0.250i)8-s + 0.929i·9-s + (−0.268 − 0.654i)10-s + 1.55·11-s + (0.0938 + 0.0938i)12-s + (−0.282 − 0.282i)13-s + 0.872i·14-s + (−0.245 + 0.100i)15-s − 0.250·16-s + (−0.266 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48416 + 1.04991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48416 + 1.04991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.06 + 0.863i)T \) |
| 43 | \( 1 + (6.45 + 1.13i)T \) |
good | 3 | \( 1 + (-0.325 + 0.325i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.30 - 2.30i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 + (1.01 + 1.01i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.10 - 1.10i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 + (-5.12 - 5.12i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 + (7.77 + 7.77i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.959T + 41T^{2} \) |
| 47 | \( 1 + (-9.36 + 9.36i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.85 + 3.85i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.97iT - 59T^{2} \) |
| 61 | \( 1 - 1.09iT - 61T^{2} \) |
| 67 | \( 1 + (8.56 - 8.56i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.6iT - 71T^{2} \) |
| 73 | \( 1 + (-5.49 + 5.49i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.25iT - 79T^{2} \) |
| 83 | \( 1 + (8.65 + 8.65i)T + 83iT^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + (5.34 - 5.34i)T - 97iT^{2} \) |
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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
−11.64374316038241137085534019404, −10.73161472347675378738543116742, −8.998570071805952872165599480323, −8.564245958207901240967125807616, −7.65132518824250346362116517895, −6.78355547367328083993184846046, −5.40865948161189685359648905228, −4.66887377576403110871784986147, −3.54824146434420074771675485227, −1.88542643413372212564558177255,
1.13158952922567215457239850917, 3.05066278490576650865385057643, 4.17304195518460219726519301090, 4.55730301768158392556592338923, 6.51032938754650244763233823596, 7.00401909628575233342209318606, 8.383996949544819930894786561609, 9.183426202140869071512766135214, 10.37406920658723585934134039332, 11.10821897061891920460479991833