| L(s) = 1 | − 2.32·3-s + 5-s + 1.39·7-s + 2.39·9-s + 4.32·13-s − 2.32·15-s + 0.601·17-s − 19-s − 3.24·21-s − 6.04·23-s + 25-s + 1.39·27-s + 4.60·29-s − 2.79·31-s + 1.39·35-s + 1.07·37-s − 10.0·39-s − 5.44·41-s + 8.64·43-s + 2.39·45-s + 1.85·47-s − 5.04·49-s − 1.39·51-s − 3.11·53-s + 2.32·57-s + 6.69·59-s − 2.64·61-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s + 0.528·7-s + 0.799·9-s + 1.19·13-s − 0.599·15-s + 0.145·17-s − 0.229·19-s − 0.708·21-s − 1.26·23-s + 0.200·25-s + 0.269·27-s + 0.854·29-s − 0.502·31-s + 0.236·35-s + 0.176·37-s − 1.60·39-s − 0.850·41-s + 1.31·43-s + 0.357·45-s + 0.269·47-s − 0.720·49-s − 0.195·51-s − 0.428·53-s + 0.307·57-s + 0.871·59-s − 0.338·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214665157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214665157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 - 0.601T + 17T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 - 1.85T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 - 6.69T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 5.59T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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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
−9.641954571447120098561244290398, −8.608736615591084490492559729062, −7.907383327387767197198710348340, −6.70313351124936683446670931635, −6.14235486214750531307888602846, −5.46480609328795584388998748613, −4.65917345370058172650245295272, −3.63028567603728988654050298155, −2.06081549124745001629732123672, −0.861758686219750748911599781021,
0.861758686219750748911599781021, 2.06081549124745001629732123672, 3.63028567603728988654050298155, 4.65917345370058172650245295272, 5.46480609328795584388998748613, 6.14235486214750531307888602846, 6.70313351124936683446670931635, 7.907383327387767197198710348340, 8.608736615591084490492559729062, 9.641954571447120098561244290398
