| L(s) = 1 | − 5.50·3-s − 5·5-s − 0.0500·7-s + 3.26·9-s − 35.1·11-s + 86.1·13-s + 27.5·15-s + 8.82·17-s − 106.·19-s + 0.275·21-s − 23·23-s + 25·25-s + 130.·27-s − 280.·29-s + 117.·31-s + 193.·33-s + 0.250·35-s + 93.3·37-s − 473.·39-s − 58.0·41-s + 508.·43-s − 16.3·45-s + 407.·47-s − 342.·49-s − 48.5·51-s + 316.·53-s + 175.·55-s + ⋯ |
| L(s) = 1 | − 1.05·3-s − 0.447·5-s − 0.00270·7-s + 0.120·9-s − 0.964·11-s + 1.83·13-s + 0.473·15-s + 0.125·17-s − 1.28·19-s + 0.00286·21-s − 0.208·23-s + 0.200·25-s + 0.930·27-s − 1.79·29-s + 0.678·31-s + 1.02·33-s + 0.00120·35-s + 0.414·37-s − 1.94·39-s − 0.221·41-s + 1.80·43-s − 0.0541·45-s + 1.26·47-s − 0.999·49-s − 0.133·51-s + 0.819·53-s + 0.431·55-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 | \( 1 + 5.50T + 27T^{2} \) |
| 7 | \( 1 + 0.0500T + 343T^{2} \) |
| 11 | \( 1 + 35.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.82T + 4.91e3T^{2} \) |
| 19 | \( 1 + 106.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 280.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 93.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 58.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 508.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 129.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 299.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 596.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 692.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 842.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 409.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−8.408760083051861632821466754392, −7.78453037656508724964930700429, −6.71679424082840866372346767257, −5.97078438200769260040028084921, −5.47744681749878706267287539350, −4.38171409601818806388279085248, −3.61947095411142783215896365997, −2.35930353363901206674365507948, −0.973825689639723361780138715666, 0,
0.973825689639723361780138715666, 2.35930353363901206674365507948, 3.61947095411142783215896365997, 4.38171409601818806388279085248, 5.47744681749878706267287539350, 5.97078438200769260040028084921, 6.71679424082840866372346767257, 7.78453037656508724964930700429, 8.408760083051861632821466754392
