| L(s) = 1 | + 1.90·3-s + 6.52·5-s + 5.22·7-s − 23.3·9-s + 21.1·11-s − 83.4·13-s + 12.4·15-s + 11.3·17-s + 7.68·19-s + 9.96·21-s + 153.·23-s − 82.3·25-s − 95.9·27-s + 29·29-s + 270.·31-s + 40.2·33-s + 34.1·35-s + 298.·37-s − 159.·39-s − 184.·41-s − 208.·43-s − 152.·45-s − 553.·47-s − 315.·49-s + 21.5·51-s + 321.·53-s + 137.·55-s + ⋯ |
| L(s) = 1 | + 0.366·3-s + 0.583·5-s + 0.282·7-s − 0.865·9-s + 0.579·11-s − 1.78·13-s + 0.214·15-s + 0.161·17-s + 0.0927·19-s + 0.103·21-s + 1.38·23-s − 0.659·25-s − 0.684·27-s + 0.185·29-s + 1.56·31-s + 0.212·33-s + 0.164·35-s + 1.32·37-s − 0.652·39-s − 0.703·41-s − 0.738·43-s − 0.505·45-s − 1.71·47-s − 0.920·49-s + 0.0592·51-s + 0.833·53-s + 0.338·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 1.90T + 27T^{2} \) |
| 5 | \( 1 - 6.52T + 125T^{2} \) |
| 7 | \( 1 - 5.22T + 343T^{2} \) |
| 11 | \( 1 - 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 83.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.68T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 321.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 104.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 464.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 745.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 509.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 0.374T + 3.89e5T^{2} \) |
| 79 | \( 1 - 610.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 342.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 601.T + 9.12e5T^{2} \) |
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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
−8.466387938717131168857234566619, −7.81383669880812795802907851896, −6.90322144254475724542664223182, −6.10858128274664950271612941829, −5.15489385806100733276816374516, −4.53369169869112976909313680943, −3.13394496298783334510622350428, −2.52629396243925500829846400147, −1.41600545151098381940867217950, 0,
1.41600545151098381940867217950, 2.52629396243925500829846400147, 3.13394496298783334510622350428, 4.53369169869112976909313680943, 5.15489385806100733276816374516, 6.10858128274664950271612941829, 6.90322144254475724542664223182, 7.81383669880812795802907851896, 8.466387938717131168857234566619
