| L(s) = 1 | − 4.43·3-s − 6.80·5-s − 24.2·7-s − 7.29·9-s − 60.0·11-s + 13·13-s + 30.2·15-s + 100.·17-s + 85.6·19-s + 107.·21-s − 63.3·23-s − 78.6·25-s + 152.·27-s + 247.·29-s + 65.4·31-s + 266.·33-s + 164.·35-s + 232.·37-s − 57.7·39-s + 47.1·41-s − 157.·43-s + 49.6·45-s + 167.·47-s + 244.·49-s − 444.·51-s − 152.·53-s + 408.·55-s + ⋯ |
| L(s) = 1 | − 0.854·3-s − 0.608·5-s − 1.30·7-s − 0.270·9-s − 1.64·11-s + 0.277·13-s + 0.520·15-s + 1.42·17-s + 1.03·19-s + 1.11·21-s − 0.574·23-s − 0.629·25-s + 1.08·27-s + 1.58·29-s + 0.379·31-s + 1.40·33-s + 0.796·35-s + 1.03·37-s − 0.236·39-s + 0.179·41-s − 0.556·43-s + 0.164·45-s + 0.518·47-s + 0.711·49-s − 1.22·51-s − 0.395·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 4.43T + 27T^{2} \) |
| 5 | \( 1 + 6.80T + 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 + 60.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 85.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 63.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 247.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 232.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 157.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 167.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 101.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 205.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 528.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 504.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 60.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 298.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 405.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 666.T + 9.12e5T^{2} \) |
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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
−8.385811027287370992588423955965, −7.83312986452642849452485145049, −6.96646554872848324061647511883, −5.93669219272967890260285607305, −5.58138050920135905958931668775, −4.54564109283152411750729775145, −3.29097341034304718662804396476, −2.78665377623433045153957580929, −0.843902399545856271215660619695, 0,
0.843902399545856271215660619695, 2.78665377623433045153957580929, 3.29097341034304718662804396476, 4.54564109283152411750729775145, 5.58138050920135905958931668775, 5.93669219272967890260285607305, 6.96646554872848324061647511883, 7.83312986452642849452485145049, 8.385811027287370992588423955965
