| L(s) = 1 | + 2.40·3-s + 5-s + 4.37·7-s + 2.76·9-s + 1.38·11-s − 2.27·13-s + 2.40·15-s + 5.99·17-s − 4.33·19-s + 10.5·21-s − 2.89·23-s + 25-s − 0.558·27-s − 8.25·29-s + 1.81·31-s + 3.32·33-s + 4.37·35-s − 1.37·37-s − 5.45·39-s + 8.97·41-s + 8.79·43-s + 2.76·45-s − 3.33·47-s + 12.1·49-s + 14.4·51-s + 10.2·53-s + 1.38·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.447·5-s + 1.65·7-s + 0.922·9-s + 0.417·11-s − 0.630·13-s + 0.620·15-s + 1.45·17-s − 0.995·19-s + 2.29·21-s − 0.603·23-s + 0.200·25-s − 0.107·27-s − 1.53·29-s + 0.325·31-s + 0.579·33-s + 0.739·35-s − 0.225·37-s − 0.873·39-s + 1.40·41-s + 1.34·43-s + 0.412·45-s − 0.486·47-s + 1.73·49-s + 2.01·51-s + 1.41·53-s + 0.186·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.703670090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.703670090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 + 8.25T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 8.97T + 41T^{2} \) |
| 43 | \( 1 - 8.79T + 43T^{2} \) |
| 47 | \( 1 + 3.33T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 + 1.58T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 1.04T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.018315851954383398294854069546, −7.72294783811864685089598278148, −6.92309712693747725584588883507, −5.75358703053255490989263214649, −5.23771232527966267110446160501, −4.19854569475537995106287986333, −3.73813536778214554284286055997, −2.48772962706259985622836437959, −2.08791080242660310979080671988, −1.16484488429923660902814085883,
1.16484488429923660902814085883, 2.08791080242660310979080671988, 2.48772962706259985622836437959, 3.73813536778214554284286055997, 4.19854569475537995106287986333, 5.23771232527966267110446160501, 5.75358703053255490989263214649, 6.92309712693747725584588883507, 7.72294783811864685089598278148, 8.018315851954383398294854069546
