| L(s) = 1 | − 5.92·3-s + 5·5-s − 24.6·7-s + 8.13·9-s + 24.6·11-s + 15.2·13-s − 29.6·15-s − 78.7·17-s + 16.3·19-s + 145.·21-s + 23·23-s + 25·25-s + 111.·27-s + 36.5·29-s − 186.·31-s − 146.·33-s − 123.·35-s + 327.·37-s − 90.4·39-s − 313.·41-s + 394.·43-s + 40.6·45-s + 252.·47-s + 263.·49-s + 466.·51-s − 8.12·53-s + 123.·55-s + ⋯ |
| L(s) = 1 | − 1.14·3-s + 0.447·5-s − 1.32·7-s + 0.301·9-s + 0.675·11-s + 0.325·13-s − 0.510·15-s − 1.12·17-s + 0.197·19-s + 1.51·21-s + 0.208·23-s + 0.200·25-s + 0.797·27-s + 0.234·29-s − 1.08·31-s − 0.770·33-s − 0.594·35-s + 1.45·37-s − 0.371·39-s − 1.19·41-s + 1.39·43-s + 0.134·45-s + 0.784·47-s + 0.768·49-s + 1.28·51-s − 0.0210·53-s + 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.92T + 27T^{2} \) |
| 7 | \( 1 + 24.6T + 343T^{2} \) |
| 11 | \( 1 - 24.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 36.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 252.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 8.12T + 1.48e5T^{2} \) |
| 59 | \( 1 + 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 420.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 658.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 144.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 521.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 987.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 176.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 769.T + 9.12e5T^{2} \) |
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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
−8.835066153492762020792064397573, −7.43168148481281232002762439858, −6.52197094940141402904200653456, −6.25088516582552885632198862171, −5.46672274368700879520266645891, −4.46156721861264489297517485710, −3.48575860692944617139826485400, −2.39412411469810094416992007657, −0.989893487603366295089609356426, 0,
0.989893487603366295089609356426, 2.39412411469810094416992007657, 3.48575860692944617139826485400, 4.46156721861264489297517485710, 5.46672274368700879520266645891, 6.25088516582552885632198862171, 6.52197094940141402904200653456, 7.43168148481281232002762439858, 8.835066153492762020792064397573
