L(s) = 1 | + (1.70 − 2.94i)2-s + 3·3-s + (−1.78 − 3.08i)4-s + 10.4·5-s + (5.10 − 8.83i)6-s + (9.10 + 15.7i)7-s + 15.0·8-s + 9·9-s + (17.7 − 30.6i)10-s + (2.36 + 4.09i)11-s + (−5.34 − 9.26i)12-s + (−44.9 + 77.8i)13-s + 61.9·14-s + 31.2·15-s + (39.9 − 69.1i)16-s + (−9.22 + 15.9i)17-s + ⋯ |
L(s) = 1 | + (0.601 − 1.04i)2-s + 0.577·3-s + (−0.222 − 0.385i)4-s + 0.931·5-s + (0.347 − 0.601i)6-s + (0.491 + 0.851i)7-s + 0.666·8-s + 0.333·9-s + (0.560 − 0.970i)10-s + (0.0647 + 0.112i)11-s + (−0.128 − 0.222i)12-s + (−0.958 + 1.66i)13-s + 1.18·14-s + 0.538·15-s + (0.623 − 1.07i)16-s + (−0.131 + 0.228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.36486 - 1.34961i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.36486 - 1.34961i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 + (543. + 74.2i)T \) |
good | 2 | \( 1 + (-1.70 + 2.94i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 + (-9.10 - 15.7i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 4.09i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (44.9 - 77.8i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (9.22 - 15.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-7.68 + 13.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-80.8 + 140. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (147. + 255. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (157. + 272. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (100. - 174. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-134. - 232. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (130. + 225. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 265.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-395. + 684. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 71 | \( 1 + (-212. - 367. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-249. + 432. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-603. - 1.04e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (632. - 1.09e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 818.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (134. - 233. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.86058480944933471641975807370, −11.21584211435572247303952977630, −9.825043059012514232369661087268, −9.296983548368535999836465078393, −7.964137720737612025759409576786, −6.60183053637595810868876955367, −5.11676608197661188304392880772, −4.09361792719878828491851382001, −2.35968107756120110690164778096, −1.98838279045922011094916336811,
1.54978889308732214865881488275, 3.39927564551611336415142590661, 5.01098581720662938991698527482, 5.63351518761328309079477874489, 7.20867422349730350222421794634, 7.56418778007559839474588199376, 8.990981113239138946848131562563, 10.19091805203812517114641237534, 10.84585727862214042509548958234, 12.71022874282666083811569980823