| L(s) = 1 | + 2i·2-s − 4·4-s + 34i·7-s − 8i·8-s − 27·11-s − 28i·13-s − 68·14-s + 16·16-s + 21i·17-s − 35·19-s − 54i·22-s + 78i·23-s + 56·26-s − 136i·28-s − 120·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.83i·7-s − 0.353i·8-s − 0.740·11-s − 0.597i·13-s − 1.29·14-s + 0.250·16-s + 0.299i·17-s − 0.422·19-s − 0.523i·22-s + 0.707i·23-s + 0.422·26-s − 0.917i·28-s − 0.768·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3425882691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3425882691\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 34iT - 343T^{2} \) |
| 11 | \( 1 + 27T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 21iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 35T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 120T + 2.43e4T^{2} \) |
| 31 | \( 1 - 182T + 2.97e4T^{2} \) |
| 37 | \( 1 + 146iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 357T + 6.89e4T^{2} \) |
| 43 | \( 1 + 148iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 84iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 702iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 840T + 2.05e5T^{2} \) |
| 61 | \( 1 + 238T + 2.26e5T^{2} \) |
| 67 | \( 1 + 461iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 708T + 3.57e5T^{2} \) |
| 73 | \( 1 + 133iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 650T + 4.93e5T^{2} \) |
| 83 | \( 1 - 903iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 735T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3iT - 9.12e5T^{2} \) |
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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
−11.35568638200711736463865723103, −10.20386161817641087865381003514, −9.268240080196071559951718991828, −8.463509399496716743683138674638, −7.80019920591031056105901603379, −6.47252091606453736172641999980, −5.61630957668637738770397233042, −4.99029791363619424145833822362, −3.32552159689396697349887064991, −2.09063581339734631606879067464,
0.11070624202348647325579644946, 1.39375735391236763612693226048, 2.92843645948722397101641318163, 4.13000755981286065308972713479, 4.81157660181719653220600379830, 6.39927166337522203123580360053, 7.38667708460933914260209778934, 8.224290522001209330590792524291, 9.431014774144783312555474101921, 10.33780407002663614960805793318
