| L(s) = 1 | − 5.59·2-s + 23.2·4-s + 5·5-s − 85.3·8-s − 27.9·10-s + 20.8·11-s − 41.4·13-s + 291.·16-s − 81.5·17-s − 15.9·19-s + 116.·20-s − 116.·22-s − 75.5·23-s + 25·25-s + 231.·26-s − 167.·29-s + 184.·31-s − 946.·32-s + 456.·34-s + 330.·37-s + 89.2·38-s − 426.·40-s + 478.·41-s − 14.5·43-s + 485.·44-s + 422.·46-s − 387.·47-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.90·4-s + 0.447·5-s − 3.77·8-s − 0.884·10-s + 0.571·11-s − 0.884·13-s + 4.55·16-s − 1.16·17-s − 0.192·19-s + 1.30·20-s − 1.12·22-s − 0.684·23-s + 0.200·25-s + 1.74·26-s − 1.07·29-s + 1.06·31-s − 5.22·32-s + 2.30·34-s + 1.46·37-s + 0.381·38-s − 1.68·40-s + 1.82·41-s − 0.0516·43-s + 1.66·44-s + 1.35·46-s − 1.20·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.59T + 8T^{2} \) |
| 11 | \( 1 - 20.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 478.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 66.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 553.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 277.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 291.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 9.11T + 5.71e5T^{2} \) |
| 89 | \( 1 + 125.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 932.T + 9.12e5T^{2} \) |
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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
−8.354390530639187101100329391789, −7.80782532473407053547668429500, −6.87161926851008155520844078204, −6.43049303036356987341918756528, −5.53232453536615591658375859492, −4.08397322951429908094444840469, −2.63627403093891440309696239381, −2.12860984251672041796697913702, −1.03354190928460682009794702401, 0,
1.03354190928460682009794702401, 2.12860984251672041796697913702, 2.63627403093891440309696239381, 4.08397322951429908094444840469, 5.53232453536615591658375859492, 6.43049303036356987341918756528, 6.87161926851008155520844078204, 7.80782532473407053547668429500, 8.354390530639187101100329391789
