L(s) = 1 | − 2.41·2-s + 3.82·4-s + 5-s − 0.828·7-s − 4.41·8-s − 2.41·10-s + 5.65·13-s + 1.99·14-s + 2.99·16-s − 1.17·17-s + 6.82·19-s + 3.82·20-s + 4·23-s + 25-s − 13.6·26-s − 3.17·28-s − 4.82·29-s + 1.58·32-s + 2.82·34-s − 0.828·35-s + 11.6·37-s − 16.4·38-s − 4.41·40-s + 4.82·41-s + 8.82·43-s − 9.65·46-s + 4·47-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.91·4-s + 0.447·5-s − 0.313·7-s − 1.56·8-s − 0.763·10-s + 1.56·13-s + 0.534·14-s + 0.749·16-s − 0.284·17-s + 1.56·19-s + 0.856·20-s + 0.834·23-s + 0.200·25-s − 2.67·26-s − 0.599·28-s − 0.896·29-s + 0.280·32-s + 0.485·34-s − 0.140·35-s + 1.91·37-s − 2.67·38-s − 0.697·40-s + 0.754·41-s + 1.34·43-s − 1.42·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.089374198\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.089374198\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.331016817388970988183371876315, −7.52826969760049550120180357149, −7.07342399351735640584013028888, −6.09585862356801641762218670768, −5.76370225568848786472270783883, −4.45812366000833001579529682874, −3.33987965375690973353145695154, −2.55876714030468699536870868796, −1.44067372505325557192109288454, −0.808556911048721832594774058780,
0.808556911048721832594774058780, 1.44067372505325557192109288454, 2.55876714030468699536870868796, 3.33987965375690973353145695154, 4.45812366000833001579529682874, 5.76370225568848786472270783883, 6.09585862356801641762218670768, 7.07342399351735640584013028888, 7.52826969760049550120180357149, 8.331016817388970988183371876315