| L(s) = 1 | − 2-s + 4-s + 0.765·5-s − 3.69·7-s − 8-s − 0.765·10-s − 5.22·11-s − 5.41·13-s + 3.69·14-s + 16-s − 1.17·19-s + 0.765·20-s + 5.22·22-s − 3.69·23-s − 4.41·25-s + 5.41·26-s − 3.69·28-s − 9.23·29-s + 1.53·31-s − 32-s − 2.82·35-s + 2.93·37-s + 1.17·38-s − 0.765·40-s + 0.317·41-s − 5.22·44-s + 3.69·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.342·5-s − 1.39·7-s − 0.353·8-s − 0.242·10-s − 1.57·11-s − 1.50·13-s + 0.987·14-s + 0.250·16-s − 0.268·19-s + 0.171·20-s + 1.11·22-s − 0.770·23-s − 0.882·25-s + 1.06·26-s − 0.698·28-s − 1.71·29-s + 0.274·31-s − 0.176·32-s − 0.478·35-s + 0.481·37-s + 0.190·38-s − 0.121·40-s + 0.0495·41-s − 0.787·44-s + 0.544·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2943706392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2943706392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 0.765T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 - 0.317T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 6.75T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.113792603933902836320856491194, −7.46406912409298490914580355973, −7.04659546092684129446429851562, −5.91519767281670043541762370794, −5.67320920191102660281936744915, −4.55668432008267924666655592995, −3.47586871213961025533574167646, −2.58054006962740798542927921989, −2.10101497365696753203686709136, −0.29745269045282451079570655365,
0.29745269045282451079570655365, 2.10101497365696753203686709136, 2.58054006962740798542927921989, 3.47586871213961025533574167646, 4.55668432008267924666655592995, 5.67320920191102660281936744915, 5.91519767281670043541762370794, 7.04659546092684129446429851562, 7.46406912409298490914580355973, 8.113792603933902836320856491194
